GIFT  OF 
rrof  .E.p  .Lewis 


Physics  Dept. 


PHYSICAL 
MEASUREMENTS 


DUFF  and  EWELL 


SCIENCl      ' 


PHYSICAL 
MEASUREMENTS 


BY 

A.  WILMER  DUFF 

PROFESSOR  OF  PHYSICS  IN  THE  WORCESTER  POLYTECHNIC  INSTITUTE 


AND 
ARTHUR  W.  EWELL 

PROFESSOR  OF  PHYSICS  IN  THE  WORCESTER  POLYTECHNIC  INSTITUTE 


THIRD  EDITION,  REVISED  AND  ENLARGED 
WITH  80  ILLUSTRATIONS 


PHILADELPHIA 

P.  BLAKISTON'S  SON  &  CO. 

1012  WALNUT  STREET 
1913 


a 


COPYRIGHT,  1913,  BY  P.  BLAKISTON'S  SON  &  Co. 
PHYSICS    DEPT. 


PREFACE  TO  THE  THIRD  EDITION. 


With  the  exception  of  the  introduction  of  a  second  method  for 
the  measurement  of  viscosity,  no  considerable  changes  will  be 
found  in  this  edition;  but  numerous  minor  improvements  have 
been  made  in  the  descriptions  of  apparatus  and  methods. 


55  7f>  J)9 


PREFACE  TO  THE  SECOND  EDITION. 


Our  intention  in  writing  this  book  was  not  to  give  an  account 
of  physical  laboratory  methods  in  general,  but  to  describe  a 
limited  number  of  carefully  chosen  exercises  such  as  we  have 
found  in  our  experience  to  be  suitable  for  the  laboratory  work  of 
students  who  have  had  a  fair  course  in  General  College  Physics. 

The  descriptions  of  the  exercises  will  usually  fit  apparatus 
and  conditions  of  considerable  diversity,  but  many  practical 
details  have  been  included  where  experience  has  shown  that  they 
are  necessary.  Other  instructors  who  may  adopt  the  book  will 
probably  find  some  of  the  exercises  unsuited  to  their  classes, 
but  the  list  is  sufficiently  extensive  to  afford  a  considerable 
variety  of  selection. 

The  descriptions  of  apparatus  are  intended  to  be  read  by  the 
student  with  the  apparatus  before  him.  Hence  elaborate  illus- 
trations have  been  thought  unnecessary.  For  an  extended 
account  of  certain  special  topics,  such  as  the  theory  of  the  balance 
and  the  construction  of  galvanometers,  references  to  other  works 
have  been  given. 

Usually  several  text-books  and  special  treatises  have  been 
referred  to  at  the  beginning  of  the  account  of  an  experiment. 
It  is  assumed  that  each  student  will  have  one  of  the  text-books 
and  that  some  of  the  special  works  will  be  found  in  the  reference 
room  of  the  laboratory.  While  the  reference  is  generally  to  the 
latest  edition  (at  the  present  date,  1910),  those  who  have  different 
editions  will  have  no  difficulty  in  finding  the  passages  referred  to. 
Each  instructor  who  uses  the  book  will  exercise  his  discretion  as 
to  what  preliminary  reading  will  be  required  and  will  issue  the 
necessary  instructions  to  his  class. 

We  are  indebted  to  Dr.  Albert  W.  Hull  for  assistance  in  reading 
the  page  proof.  Many  of  the  tables  have  been  taken  from  Ewell's 
Physical  Chemistry. 


vii 


CONTENTS. 


PAGE 

CTENERAL  INTRODUCTION i 

I.  Purpose  of  Course.  2.  General  Directions.  3.  Reports. 
4.  Errors.  5.  Errors  of  Observation.  6.  Possible  Error 
of  a  Calculated  Result.  7.  General  Method  for  the  Possible 
Error  of  a  Result.  8.  Some  General  Notes  on  Errors.  9. 
Probable  Error  of  a  Mean.  10.  Limits  to  Calculations.  II. 
Notation  of  Large  and  Small  quantities.  12.  Plotting  of 
Curves. 

MECHANICS 12 

13.  The  Use  of  a  Vernier.  14.  Vernier  Caliper.  15.  Microm- 
eter Caliper.  16.  Micrometer  Microscope.  17.  Com- 
parator. 1 8.  Sphere  meter.  19.  Dividing  Engine.  20. 
Cathetometer.  21.  Barometer.  22.  The  Balance.  23.  Ad- 
justment of  Telescope  and  Scale.  24.  Time  Determination. 

I.  To  Make  and  Calibrate  a  Scale. 
II.  Errors  of  Weights. 

III.  Volume,  Mass,  and  Density  of  a  Regular  Solid. 

IV.  Mohr-Westphal  Specific  Gravity  Balance. 
V.  Density  by  the  Volumenometer. 

VI.  Density  of  Gases. 

VII.  Acceleration  of  Gravity  by  Pendulum. 
VIII.  Coefficient  of  Friction. 
IX.  Hooke's  Law  and  Young's  Modulus. 
X.  Rigidity  (or  Shear  Modulus). 
XI.  Viscosity. 
XII.  Surface  Tension. 

HEAT 59 

25.  Radiation  Correction  in  Calorimetry.  26.  The  Beck- 
mann  Thermometer. 

XIII.  Thermometer  Testing. 

XIV.  Temperature  Coefficient  of  Expansion. 

XV.  Coefficient  of  Apparent  Expansion  of  a  Liquid. 
XVI.  Coefficient  of  Increase  of  Pressure  of  Air. 
XVII.  Pressure  of  Saturated  Water  Vapor. 

ix 


X  CONTENTS. 

PAGE 

XVIII.  Hygrometry. 

XIX.  Specific  Heat  by  Method  of  Mixture. 

XX.  Ratio  of  Specific  Heats  of  Gases. 

XXI.  Latent  Heat  of  Fusion. 

XXII.  Latent  Heat  of  Vaporization. 

XXIII.  Latent  Heat  of  Vaporization.     Continuous-flow  method. 

XXIV.  Thermal  Conductivity. 

XXV.  The  Mechanical  Equivalent  of  Heat. 
XXVI.  The  Melting-point  of  an  Alloy. 
XXVII.  Heat  Value  of  a  Solid. 
XXVIII.  Heat  Value  of  a  Gas  or  Liquid. 
XXIX.  Pyrometry. 

SOUND in 

XXX.  The  Velocity  of  Sound. 
XXXI.  The  Velocity  of  Sound  by  Kundt's  Method. 

LIGHT 116 

27.  Monochromatic  Light.  28.  Rule  of  Signs  for  Spherical 
Mirrors  and  Lenses. 

XXXII.  Photometry. 

XXXIII.  Spectrometer  Measurements. 

XXXIV.  Radius  of  Curvature. 
XXXV.  Focal  Length  of  a  Lens. 

XXXVI.  Lens  Combinations. 
XXXVII.  Magnifying  Power  of  a  Telescope. 
XXXVIII.  Resolving  Power  of  Optical  Instruments. 
XXXIX.  Wave-length  of  Light  by  Diffraction  Grating. 

XL.  Interferometer. 
XLI.  Rotation  of  Plane  of  Polarization. 

ELECTRICITY  AND  MAGNETISM 144 

29.  Resistance  Boxes.  30.  Forms  of  Wheatstone's  Bridge. 
31.  Galvanometers.  32.  Correction  for  Damping  of  a  Bal- 
listic Galvanometer.  33.  Galvanometer  Shunts.  34. 
Standard  Cells.  35.  Device  for  Getting  a  Small  E.  M.  F. 
36.  Double  Commutator.  37.  Relations  between  Electrical 
Units. 

XLII.  Horizontal  Component  of  the  Earth's  Magnetic  Field. 
XLIII.  Magnetic  Inclination  or  Dip. 

XLIV.  Measurement  of  Resistance  by  Wheatstone's  Bridge. 
XLV.  Galvanometer  Resistance  by  Shunt  Method. 
XLVI.  Galvanometer  Resistance  by  Thomson's  Method. 
XLVII.  Measurement  of  High  Resistance  (i). 


CONTENTS.  xi 


XLVIII.  Measurement  of  High  Resistance  (2). 
XLIX.  Measurement  of  Low  Resistance  (i). 
L.  Measurement  of  Low  Resistance  (2). 
LI.  Measurement  of  Low  Resistance  (3). 
LI  I.  Comparison  of  Resistances  by  the  Carey-Foster  Method. 
LI II.  Battery  Resistance  by  Mance's  Method. 
LIV.  Temperature  Coefficient  of  Resistance. 
LV.  Specific  Resistance  of  an  Electrolyte. 
LVI.  Comparison  of  E.  M.  F.'s  by  High  Resistance  Method. 
LVII.  Comparison  of  E.  M.  F.'s  and  Measurement  of  Battery 

Resistance  by  Condenser  Method. 
LVI  1 1.  Measurement  of  Potential  Differences  by  Potentiometer 

Method. 

LIX.  Measurement  of  Current  by  Potentiometer  Method. 
LX.  Comparison  of  Capacities  of  Condensers. 
LXI.  Absolute  Determination  of  Capacity. 
LXII.  Coefficients  of  Self-induction  and  of  Mutual  Induction. 
LXIII.  Strength  of  a  Magnetic  Field  by  a  Bismuth  Spiral. 
LXIV.  Study  of  a  Ballistic  Galvanometer. 
LXV.  Magnetic  Permeability. 
LXVI.  Magnetic  Hysteresis. 
T  XVTT     /    ^   Mechanical  Equivalent  of  Heat. 

\    (b)  Horizontal  Intensity  of  Earth's  Magnetism. 
LXVI II.  Thermoelectric  Currents. 
LXIX.  Elementary    Study   of    Resistance,    Self-induction,    and 

Capacity. 

LXX.  Self-induction,  Mutual  Induction,  and  Capacity,  Alter- 
nating Currents. 

LXXI.  Dielectric  Constant  of  Liquids. 
LXXII.  Electric  Waves  on  Wires. 


TABLES 222 

I.  Four- Place  Logarithms. 
II.  Trigo metrical  Functions. 

III.  Reduction  to  Infinitely  Small  Arc. 

IV.  Barometer  Corrections. 

V.  Density  of  Specific  Volume  of  Water. 
VI.  Density  of  Gases. 
VII.  Density,  Specific  Heat,  and  Coefficient  of  Expansion  of 

Metals. 
VIII.  Density,  Specific  Heat,  and  Coefficient  of  Expansion  of 

Miscellaneous  Substances. 
XI.  Elastic  Moduli. 
X.  Surface  Tension. 


xii  CONTENTS. 

PAGE 

XL  Coefficient  of  Viscosity. 

XII.  Specific  Heats  of  Gases. 

XIII.  Vapor  Pressure  of  Water. 

XIV.  Boiling-point  of  Water. 

XV.  Wet  and  Dry  Bulb  Hygrometer. 
XVI.  Vapor  Pressure  of  Mercury. 
XVII.  Melting-points  of  Metals. 
XVIII.  Wave-lengths  of  Light. 
XIX.  Refractive  Indices. 
XX.  Specific  Rotatory  Power. 
XXL  Photometric  Table. 

XXII.  Specific    Resistance    and    Temperature    Coefficient    of 
Metals. 

XXIII.  Specific  Resistance  and  Temperature  Coefficient  of  Solu- 

tions. 

XXIV.  Dielectric  Constants. 

INDEX  .  .  241 


PHYSICAL  MEASUREMENTS. 


INTRODUCTION. 


1.  Purpose  of  Course. 

Intelligent  work  requires  a  clear  perception  of  the  end  in 
view.  It  is  important,  therefore,  to  remember  that  the  purpose 
of  a  course  in  Laboratory  Physics  is  not  only  the  attainment, 
by  personal  experimentation,  of  a  more  definite  knowledge  of  the 
facts  and  principles  of  physics  and  an  acquaintance  with  the  use 
of  measuring  instruments  and  methods,  but  also  the  acquisition 
of  a  scientific  habit  of  accuracy  and  carefulness  in  observing  and 
examining  phenomena  and  drawing  conclusions  therefrom. 

2.  General  Directions. 

Much  time  in  the  laboratory  will  be  wasted  unless  some  prep- 
aration be  made  before  coming  to  the  laboratory.  The  purpose 
and  general  method  of  the  measurement  to  be  made  should  be 
studied  with  the  aid  of  this  text-book  and  one  of  the  text-books 
of  General  Physics  mentioned  in  the  "Elementary  References" 
preceding  the  directions.*  This  may  usually  be  done  in  a  few 

*  These  elementary  text  books  (one  of  which  has  probably  been  studied  by 
the  student  in  an  earlier  course  and  is  in  his  possession)  are  for  brevity  desig- 
nated by  the  names  of  their  authors  as  follows: 

Ames  refers  to  Ames'  Text-book  of  General  Physics. 

Crew  refers  to  Crew's  General  Physics. 

Duff  refers  to  Text-book  of  Physics. 

Edser  (Heat)  refers  to  Edser's  Heat  for  Advanced  Students. 

Edser  (Light)  refers  to  Edser's  Light  for  Students. 

Hadley  refers  to  Hadley's  Magnetism  and  Electricity  for  Students. 

Kimball  refers  to  Kimball's  College  Physics. 

Reed  &  Guthe  refers  to  Reed  &  Guthe's  College  Physics. 

Spinney  refers  to  Spinney's  Text-book  of  Physics. 

Watson  refers  to  Watson's  Text-book  of  Physics. 

Watson  (Pr)  refers  to  Watson's  Text-book  of  Practical  Physics. 

Other  books,  journals  or  memoirs  are  usually  referred  to  by  their  full  titles. 


2  INTRODUCTION. 

minutes  at  home,  whereas  it  might  require  an  hour  or  more  in  a 
laboratory  where  a  number  of  people  are  moving  around. 

The  readings  made  in  the  laboratory  should  always  be  recorded 
in  a  firmly  bound  book  reserved  for  this  purpose  only,  and  never 
on  loose  slips  of  paper  or  in  a  book  that  may  become  dog-eared 
and  untidy.  When,  for  convenience  or  of  necessity,  two  work 
together  at  an  experiment,  each  should  keep  his  own  notes  of  the 
measurements  made,  and,  whenever  possible,  each  should  make 
a  separate  set  of  readings  for  himself,  and  these  should  be  as 
independent  as  possible. 

No  operation  should  be  performed  or  measurement  made 
unless  the  purpose  and  meaning  of  it  are  understood ;  otherwise 
it  may  be  made  imperfectly  or  some  essential  part  of  it  may  be 
overlooked. 

3.  Reports. 

An  essential  part  of  the  work  is  a  written  report  on  each  experi- 
ment completed.  This  should  be  handed  in  within  a  week  after 
the  work  is  finished.  In  preparing  the  report  the  writer  has  to 
make  clear  to  himself  the  purpose  and  bearing  of  each  part  of  the 
work  and  examine  critically  the  value  and  accuracy  of  the  final 
result.  This  exercise  is  as  valuable  as  the  experimental  work 
itself.  The  report  should  be  as  brief  as  possible,  consistently 
with  giving  the  following  information : 

The  purpose  of  the  experiment  (including  the  definition  of  the 
leading  terms,  such  as  coefficient  of  friction,  mechanical  equiva- 
lent of  heat,  etc.) ; 

A  brief  statement  of  the  method  used ; 

A  statement  (tabulated  if  possible)  of  the  observations  and 
readings  made : 

An  outline  of  the  calculation  of  the  final  result  (omitting  the 
details  of  the  numerical  work) ; 

A  criticism  of  the  reliability  of  the  result; 

Brief  answers  to  the  questions  appended  to  the  directions. 

4.  Errors. 

A  perfectly  accurate  experimental  result  is  impossible;  but 
some  estimate  can  usually  be  formed  as  to  the  magnitude  of  the 


ERRORS   OF   OBSERVATION.  3 

possible  error  and  this  is  frequently  of  the  greatest  value.  An 
experimental  result  of  unknown  reliability  is  often  of  very  little 
value.  Hence  an  estimate  of  the  accuracy  of  a  measurement  is 
very  desirable  in  an  account  of  the  work. 

Inaccuracy  may  arise  from  several  different  causes — (i)  errors 
of  observation,  due  to  the  inherent  limitations  of  the  observer's 
powers  of  observing  and  judging;  (2)  instrumental  errors,  arising 
from  imperfections  in  the  work  of  the  instrument  maker  in  con- 
structing and  subdividing  the  scale  used  by  the  observer;  (3) 
mistakes,  such  as  the  mistaking  of  an  8  for  a  3  on  a  scale;  (4) 
systematic  errors  due  to  faultiness  in  the  general  method  employed. 

Instrumental  errors  may  be  decreased  by  using  more  accurate 
instruments  or  by  calibrating  the  scales  of  the  instruments  used, 
that  is,  ascertaining  and  allowing  for  the  errors  in  their  gradua- 
tion. This  is  frequently  a  difficult  operation  and  unsuited  for  an 
elementary  course.  When  the  contrary  is  not  stated  we  shall 
assume  that  the  accuracy  of  the  instruments  is  such  that  the 
instrumental  errors  are  less  than  the  errors  of  observation. 

Mistakes  in  reading  can  be  eliminated  by  care  and  repetition. 
Systematic  errors  are  apt  to  arise  when  some  indirect  method  of 
arriving  at  a  result  is  adopted,  a  direct  method  being  difficult 
or  impossible.  For  example,  the  length  of  a  wave  of  light  cannot 
be  measured  directly  and  a  method  depending  on  diffraction  or 
interference  is  usually  employed  (Exp.  XXXIX).  A  careful 
study  of  the  method  used  will  often  enable  us  to  eliminate  such 
errors  by  improving  the  details  of  the  method,  or,  where  this  can- 
not be  done,  some  estimate  of  the  uneliminated  errors  can  often 
be  formed. 

5.  Errors  of  Observation. 

Different  methods  of  estimating  the  magnitude  of  errors  of 
observation  may  be  employed,  the  choice  depending  on  the  nature 
of  the  measurements.  In  many  cases  the  quantity  can  be  meas- 
ured several  times  and  the  mean  taken,  it  being  probably  more 
accurate  than  a  single  observation.  In  other  cases  circumstances 
do  not  permit  repetition  and  a  single  observation  must  suffice. 
In  either  case  the  observer  can,  from  the  circumstances  of  the 
case,  say  with  a  high  degree  of  probability  that  the  error  cannot 


4  INTRODUCTION. 

be  greater  than  a  certain  magnitude.  This  we  shall  call  the 
" possible  error"  of  the  measurement.  It  does  not  strictly  mean 
the  greatest  possible  error,  since  a  greater  error  might  be  theo- 
retically possible  but  very  improbable. 

(a)  When  Only  a  Single  Observation  is  Made. — For  example, 
a  liquid,  the  temperature  of  which  is  varying  slowly,  is  kept  well 
stirred  and  the  temperature  is  observed  by  means  of  a  ther- 
mometer graduated  to  degrees.     The  temperature  at  a  certain 
time  is  noted  as  being  between  36°  and  37°  and  the  observer, 
estimating  to  o.i  of  a  division,  records  the  temperature  as  36.3°; 
but  he  does  not  trust  his  estimate  closer  than  o.i;    that  is,  he 
considers  that  the  real  temperature  may  be  as  high  as  36.4°  or 
as  low  as  36.2°.     He  therefore  states  the  temperature  as  36.3° 
with  a  possible  error  of  0.1°,  or  36.3°  =*=  0.1°.     The  actual  error 
may,  of  course,  be  less  than  0.1°;   the  latter  is  only  a  reasonable 
estimate  of  the  limit  of  error  of  observation. 

(b)  When  Several  Different  Observations  of  a  Quantity  are  Made. 
—The  mean  of  a  number  of  observations  of  a  quantity  is  more 

trustworthy  than  a  single  reading,  for  observations  that  are  too 
large  are  likely  to  counterbalance  others  that  are  too  small. 
Greater  confidence  can  be  placed  in  the  mean  when  the  separate 
readings  differ  but  little  from  the  mean  than  when  they  differ 
greatly.  The  average  of  the  differences  between  the  mean  and 
the  separate  readings  is  called  the  mean  deviation.  It  can  be 
shown  (as  indicated  in  §9)  that  when  ten  observations  are  made, 
the  probability  that  the  actual  error  is  greater  than  the  mean 
deviation  is  very  small,  about  I  in  100,  while  if  15  observations 
are  made  it  is  reduced  to  I  in  1000.  Even  if  only  5  observations 
are  made  (which  is  rather  too  small  a  number)  the  probability 
is  only  I  in  15.  Hence,  when  a  quantity  is  measured  several 
times,  the  average  deviation  may  be  taken  as  a  measure  of  the 
possible  error. 

6.  Possible  Error  of  a  Calculated  Result. 

A  piece  of  laboratory  work  usually  calls  for  the  measurement 
of  several  different  quantities  and  the  calculation  of  a  result  by 
some  formula.  Knowing  the  possible  errors  of  the  separate 
quantities  we  can  deduce  the  possible  error  of  the  result,  but  the 


POSSIBLE   ERROR   OF  A   CALCULATED   RESULT.  5 

method  will  vary  with   the  nature  of  the  arithmetical  opera- 
tions. 

(a)  Possible  Error  of  a  Sum  or  Difference. — The  possible  error 
of  a  sum  or  difference  is  the  sum  of  the  possible  errors  of  the 
separate  quantities,  for  each  possible  error  may  be  either  positive 
or  negative. 

Example. — A  bulb  containing  air  (Exp.  VI)  weighs  20.1425  g. 
±0.0002  g.  and  after  the  air  has  been  pumped  out  it  weighs 
20.0105  g.  ±0.0002  g.  Hence  the  weight  of  the  air  is  0.1320  g. 
±0.0004  g.  Since  it  is  sometimes  erroneously  assumed  that  a 
derived  result  must  be  accurate  to  as  high  a  percentage  as  the 
measurements  from  which  it  is  deduced,  it  should  be  noticed  in 
the  above  that,  while  the  separate  weights  are  found  to  0.001%, 
the  weight  of  the  air  is  only  ascertained  to  0.3%. 

(b)  Possible  Error  of  a  Power. — If  a  measured  quantity  x  is 
in  doubt  by  p  per  cent  (p  being  small),  the  nth  power  of  x  is  in 
doubt  by  np  per  cent.     For 


IOO 

squares  and  higher  powers  of  p/ioo  being  neglected. 

Example  of  (a)  and  (b). 

T  =  3.506  ±  .005  and  /  =  2.018  ±  .003.  (Exp.  X).  What 
is  the  possible  error  of  T2 — t2?  T2-  =  12.29  and  since  T  may  be 
in  error  by  1/7%,  T2  may  be  in  error  by  2/7%  or  .04.  Hence 
T2  =  12.29  =*=  -04.  Similarly  t2  =  4.07  ±  .01.  Hence  T2 — 12  = 

8.22   ±   .05. 

(c)  Possible  Error  of  a  Product  or  Quotient. — The  percentage 
by  which  a  product  or  quotient  is  in  doubt  is  the  sum  of  the  per- 
centages by  which  the  separate  quantities  are  in  doubt.  For 
if  the  quantities  be 

x{  I  ±~—  I  and  y{  I  ±-5 --  I 
\       ioo/  V       ioo/ 

their  product  is 


loo  loo  loo 


6  INTRODUCTION. 

and  their  quotient  is 


p/ioo  and  q/ioo  being  assumed  small.  It  is  evident  that  a 
similar  statement  applies  to  any  number  of  products  and  quo- 
tients. 

Example  of  (b)  and  (c). 

The  diameter  of  a  sphere  (Exp.  Ill)  is  measured  by  a  vernier 
caliper  and  found  to  be  1.586  cm.,  but  the  vernier  only  reads  to 
1/50  mm.;  so  the  possible  error  is  .002  cm.  or  1/8  of  i%.  The 
sphere  is  weighed  in  a  balance  such  that  I  mg.  added  to  one  pan 
does  not  cause  an  observable  change  of  the  pointer,  while  2  mg. 
does,  and  the  weight  is,  therefore,  16.344  g-  with  a  possible  error 
of  .002  g.  or  1/80%.  The  calculated  value  of  the  density  is 
7.827;  but  the  volume  may  be  in  error  by  3/8%  and  the  mass  by 
1/80%.  Hence  the  density  may  be  in  error  by  3/8  +  1/80%,  or 
practically  3/8%.  Hence  the  proper  statement  of  the  density 
is  7.83  with  a  possible  error  of  .03  or  7.83  =*=  .03. 

7.  General  Method  for  the  Possible  Error  of  a  Result. 

The  above  rules  for  sums,  differences,  powers,  products,  and 
quotients  will  usually  suffice  for  finding  the  possible  error  of  a 
result  calculated  from  the  measurements  of  several  quantities. 
But  when  several  of  these  operations  are  combined,  or  when  the 
formula  for  calculation  contains  one  of  the  quantities  more  than 
once,  the  effects  of  the  several  errors  may  be  difficult  to  trace  by 
these  means.  The  following  general  method  is  always  applicable. 
It  may  be  carried  out  by  simple  arithmetic,  but  is  simplified  by 
an  elementary  use  of  the  calculus. 

To  find  to  what  extent  the  possible  error  in  one  of  the  quanti- 
ties affects  the  result,  we  may  calculate  the  result  assuming  all 
the  quantities  to  be  quite  accurate  and  then  repeat  the  calcula- 
tion after  changing  one  of  the  quantities  by  its  possible  error. 
The  difference  in  the  result  will  be  the  effect  sought.  It  we  do  the 
same  for  each  of  the  other  quantities,  the  final  possible  error  of 
the  result  will  be  the  sum  (without  regard  to  sign)  of  the  parts 
due  to  the  separate  quantities. 


GENERAL   METHOD   FOR   THE   POSSIBLE   ERROR.  7 

This,  however,  is  equivalent  to  differentiating  the  whole  ex- 
pression, first  with  regard  to  one  quantity,  then  with  regard  to  a 
second  and  so  on  and  finally  adding  the  partial  differentials.  It 
will  be  seen  from  the  following  examples  that  the  process  is  much 
simplified  by  taking  the  logarithm  of  the  whole  formula  before 
differentiating. 

(i)  Time  of  Vibration  of  a  Pendulum  (Exp.  VII).  —  If  in  time 
T  a  pendulum  makes  n  fewer  vibrations  than  the  pendulum  of  a 
clock  that  beats  seconds  and  if  /  is  the  time  of  a  single  virbation, 

t-  T 
—f=z 

Taking  logarithms, 

log  t  =  log  T-log  (T-n) 
Hence  by  differentiating, 

dt  =  dT       dT 
t~f     T-n 


T(T-n) 

This  means  that  if  T  be  changed  by  a  small  quantity,  dT,  the 
consequent  change,  dt,  in  t  is  given  by  the  formula.  If  the  pos- 
sible error  of  T  be  2  seconds,  by  putting  dT  =  =*=  2  the  value  of 
dt  will  be  the  possible  error  of  /.  If  T  be  862  seconds  and  n  be  17, 


This  indicates  one  of  the  advantages  of  taking  logarithms.  It 
gives  us  at  once  the  ratio  of  5t  to  /,  or  (multiplied  by  100)  the 
percentage  by  which  /  is  in  doubt. 

(2)  Specific  Heat  by  the  Method  of  Mixture  (Exp.  XIX).— 
Let  T  =  95°  be  the  initial  temperature  of  ihe  specimen,  /0  =  25° 
that  of  the  water,  and  let  /  =  45°  be  the  final  temperature  of  the 
mixture,  and  let  the  possible  error  of  each  thermometer  reading 
be  0.2°.  The  formula  for  calculation  is 


M(T-t) 


8  INTRODUCTION. 

We  shall  consider  how  far  the  possible  errors  in  the  thermometer 
readings  affect  x,  leaving  the  consideration  of  the  other  terms 
(the  errors  of  which  are  likely  to  be  much  smaller)  to  the  reader. 

log  x  =  log  (m  +  mis)  +  log  (t  —  t^—log  M—log  (T—t) 

Proceeding  as  in  (i)  above,  we  find  the  effects  of  the  possible 
errors  of  T,  /o,  and  /  respectively  as  follows  : 

dx  5T  0.2 

*  =  ~T-t  50            :°4% 

dx_  dtp  0.2 

X  ~  t  —  t0  20 

dx  _     (T-tp)  dt  70 

°': 


Total  =  2.8% 

This  example  will  show  a  second  advantage  in  the  method  of 
taking  logarithms.  It  separates  the  various  terms  and  so  simpli- 
fies the  process. 

8.  Some  General  Notes  on  Errors. 

The  statement  of  a  possible  error  should  contain  only  one 
significant  figure.  (A  zero  that  serves  only  to  fix  the  decimal 
point,  such  as  the  zeros  in  0.0026,  is  not  a  significant  figure). 
Thus  in  the  last  example  in  §6,  3/8%  of  7.83  is  0.0293,  which 
shows  that  the  second  decimal  place  in  7.83  is  in  doubt  by  3. 
Hence  it  would  be  superfluous  to  add  figures  to  show  that  the 
third  and  fourth  decimal  places  are  also  in  doubt. 

Measurements  sometimes  seem  so  accurate  that  one  is  tempted 
to  say  that  "the  possible  error  is  practically  zero  and  need  not  be 
considered."  This  is  never  literally  true.  One  factor  may  be  so 
accurately  determined,  compared  with  other  factors,  that  the 
effect  of  its  possible  error  on  the  result  might  seem  to  be  negligible  ; 
but  only  a  calculation  can  show  this  and  the  calculation  will 
frequently  show  the  opposite.  An  illustration  occurs  in  the  first 
example  of  §6,  considered  in  connection  with  the  other  measure- 
ments required  to  determine  the  density  of  air  in  Exp.  III. 

The  consideration  of  possible  errors  is  of  great  importance  in 
deciding  what  care  need  be  expended  in  determining  the  various 


PROBABLE    ERROR       OF   A   MEAN.  9 

factors  in  a  complex  measurement  and  what  are  the  best  condi- 
tions for  obtaining  an  accurate  result.  -  This  applies  more 
especially  to  advanced  and  difficult  measurements,  but  illustra- 
tions will  occur  in  this  book.  (Exp.  XIX.)  But,  as  one  of  the 
purposes  of  this  course  is  to  teach  the  most  exact  use  of  the 
measuring  instruments,  measurements  should  usually  be  made 
as  accurately  as  the  instruments  will  permit. 

9.  "Probable  Error"  of  a  Mean. 

There  is  another  method  of  indicating  the  reliability  of  meas- 
urements which  possesses  some  advantages  over  the  one  that  we 
have  explained,  though  it  is  not  so  generally  applicable.  When 
a  large  number  of  observations  of  a  quantity  have  been  made, 
we  can,  by  means  of  formulas  deduced  from  the  mathematical 
Theory  of  Probability,  calculate  the  probability  that  the  mean 
is  not  in  error  by  more  than  a  given  amount.  When  a  coin  is 
tossed  up  it  is  an  even  chance  whether  it  will  come  down  a  head 
or  a  tail ;  the  chance  or  probability  of  its  being  a  head  is,  therefore, 
i  in  2  or  1/2.  Now  the  "probable  error"  of  the  mean  of  a  num- 
ber of  readings  is  denned  as  a  magnitude  such  that  it  is  an  even 
chance  whether  the  error  is  greater  or  whether  it  is  less  than  this 
magnitude.  In  other  words,  the  probability  of  the  error  exceed- 
ing the  "probable  error"  is  1/2.  One  formula  for  calculating  the 
"probable  error"  is  the  following: 

average  deviation 

£  =  0.84-7—  ==  ==±. 

V number  of  observations 

This  method  is  useful  as  a  method  of  indicating  the  reliability 
of  measurements  when  each  of  all  the  quantities  that  occur  in 
the  experiment  can  be  measured  several  times.  For  when  each 
mean  and  its  "probable  error"  has  been  found  we  can  calculate 
the  "probable  error"  of  the  final  result.  For  further  details  we 
shall  refer  the  reader  to  other  works  (e.  g.,  Merriman's  "Least 
Squares").  We  shall  not  have  frequent  occasion  to  refer  to 
"probable  errors,"  since  in  most  cases  some  of  the  quantities 
that  have  to  be  determined  cannot  be  measured  more  than  once. 

In  justification  of  the  use  of  the  mean  deviation  as  a  measure 
of  the  possible  error,  we  may  note  that  by  the  above  formula  for 


IO  INTRODUCTION. 

e,  when  10  observations  have  been  made,  the  mean  deviation 
equals  3.6  e.  Now  a  reference  to  tables  of  the  probability  of 
errors  (e.  g.,  Smithsonian  Tables)  shows  that  the  probability 
of  an  error  greater  than  3.8  e  is  about  I  in  100. 

10.  Limits  to  Calculations. 

By  the  above  methods  the  possible  error  in  any  calculation 
from  experimental  quantities  may  be  deduced.  The  magnitude 
of  the  possible  error  in  any  calculation  indicates  how  far  it  is 
useful  and  desirable  to  carry  the  calculation.  A  calculation 
should  be  carried  as  far  as,  but  not  farther  than,  the  first  doubtful 
figure.  This  rule  must  be  applied  not  only  to  the  calculation  of 
the  final  result,  but  also  to  each  intermediate  step.  When  a 
calculation  is  carried  too  far,  useless  and  very  unscientific  labor 
is  expended,  and  when  it  is  not  carried  far  enough  very  absurd 
results  are  often  obtained. 

In  addition  and  subtraction  a  place  of  decimals  that  is  doubtful 
in  any  one  of  the  quantities  is  doubtful  in  the  result. 

In  multiplication  and  division  (performed  in  the  ordinary  way) 
decimal  places  that  have  not  been  determined  are  usually  filled 
up  by  zeros.  Any  figure  in  the  result  that  would  be  altered  by 
changing  one  of  these  zeros  to  5  is  doubtful. 

ii.  Notation  of  Very  Large  and  Very  Small  Numbers. 
Partly  to  save  space  and  partly  to  indicate  at  once  the  magni- 
tude of  very  large  or  very  small  numbers,  the  following  notation 
is  used.  The  digits  are  written  down  and  a  decimal  point  placed 
after  the  first  and  its  position  in  the  scale  indicated  by  multi- 
plying by  some  power  of  10.  Thus  42140000  is  written  4.2 14  X 
io7  and  .00000588  is  written  5.88  X  io~6.  This  also  enables  us 
to  abbreviate  the  multiplication  and  division  of  such  numbers. 
Thus  42140000  X  .00000588  is  the  same  as  4.214  X  5.88  X  io 
and  42140000  -T-  .00000588  is  the  same  as  (4.214  ^  5.88)  X  io13. 

12.  Plotting  of  Curves. 

It  is  assumed  that  the  general  method  of  the  representation  of 
the  connection  between  two  related  quantities  by  means  of  a 


PLOTTING   OF   CURVES.  II 

curve  is  familiar  to  the  reader  from  his  work  in  Graphical  Algebra 
or  elsewhere.     Attention  may  be  called  to  the  following  points: 

1.  Mark  experimental  points  clearly  by  crosses  or  circles  sur- 
rounding the  points. 

2.  The  curve  should  be  drawn  so  as  to  strike  an  average  path 
among  the  points ;  it  does  not  have  to  pass  through  even  one  point. 

3.  Abscissas  may  be  drawn  to  any  scale  and  ordinates  to  any 
scale.     Record  the  main  division  of  the  scales  along  each  axis. 
(Do  not  record  the  individual  observations  on  the  axes.) 

4.  For  convenience,  all  the  abscissas  may  be  diminished  by  the 
same  amount  before  plotting,  and  the  same  is  true  of  ordinates. 

5.  Ordinates  and  abscissas  should  be  drawn  to  such  scales  that 
the  curve  occupies  a  large  part  of  the  paper. 

6.  Curves  should  be  drawn  carefully  and  neatly  by  means  of 
curve  forms. 


MECHANICS. 
13.  The  Use  of  a  Vernier. 

The  vernier  is  a  contrivance  for  reading  to  fractions  of  the  units 
in  which  scales  are  graduated.  It  is  a  second  scale  parallel  to 
the  main  scale  of  the  instrument  and  so  divided  that  n  of  its 
units  equal  n—  i  or  n  +  I  of  the  units  of  the  scale.  If  5  is  the 
length  of  a  scale  unit  and  v  that  of  a  vernier  unit,  in  the  first  case 

n  v  =  (n  —  i)sors  —  v  =  s-^-n; 

in  the  second  case 

n  v  =  (n  -}-  i)  s  or  v  —  s  =  s  -r-  n. 

Hence  the  unit  of  the  vernier  is  less  or  greater  than  that  of  the 
scale  by  one-wth  of  a  scale  unit. 

If  we  did  not  have  a  vernier  there  would  be  something  in  the 
nature  of  an  index  to  indicate  what  division  of  the  scale  should  be 
read  in  making  a  certain  measurement  and  fractions  would  be 
estimated  by  eye.  The  zero  of  the  vernier  is  taken  as  such  an 
index,  the  whole  number  of  scale  divisions  being  the  number  just 

.        below  the  zero  of  the  vernier,  while 
\      the  fraction  of  a  scale  division  is  de- 


I  I  I.I  I  I  I  I  I  I  MM  II  \    termined  with  the  vernier.     If  the 
A|0'  '  *  M  *       Uo  mth  division  of  the  vernier  coincides 

^  •  with  the  scale  division,  the  zero  of 

rIG.  I.  .  ir 

the  vernier  must  be  m  nths  ot  a 

scale  unit  from  the  scale  division  just  below  it.  Thus  the  use  of 
a  vernier  divided  into  n  parts  is  equivalent  to  subdividing  the 
scale  unit  into  n  parts. 

If  no  vernier  division  exactly  coincides  with  a  scale  division 
there  will  be  two  vernier  divisions  nearly  coincident  with  scale 
divisions,  and  one  can  often  estimate  fractions  of  the  fraction 

12 


MICROMETER  CALIPER.  13 

given  by  the  vernier.  In  Fig.  I  the  whole  number  of  scale  divi- 
sions is  5.2,  and  evidently  the  third  and  fourth 
vernier  divisions  most  nearly  coincide  with  scale 
divisions.  Since  the  third  division  is  somewhat 
nearer  a  coincidence,  we  may  call  the  fraction 
3.3  tenths,  or,  the  full  reading  will  be  5.233. 


14.  Vernier  Caliper. 

The  vernier  caliper  consists  of  a  straight  gradu- 
ated bar,  and  two  jaws  at  right  angles  to  it,  one 
of  which  is  fixed  while  the  other  is  movable. 
The  position  of  the  movable  jaw  can  be  accurately 
determined  by  means  of  the  scale  and  a  vernier 
which  should  read  zero  when  the  jaws  are  in 
contact.  (If  this  be  not  the  case,  allowance  must  be  made  for 
the  zero  reading). 

15.  Micrometer  Caliper. 

The  micrometer  caliper  is  a  U-shaped  piece  of  metal  in  one  arm 
of  which  is  a  steel  plug  with  a  carefully  planed  face  and  through 
the  other  arm  of  which  passes  a  screw  with  a  plane  end  parallel 
and  opposite  to  that  of  the  screw.  A  linear  scale  on 
the  frame  reads  zero  approximately  when  the  plug 
and  screw  are  in  contact,  and  its  reading  in  any 
other  position  indicates  the  whole  number  of  turns 
of  the  screw  and  consequently  the  number  of  mm. 
(or  1/2  mm.  or  1/40  in.  as  the  pitch  of  the  screw  may 
be)  between  the  screw  and  the  plug.  Fractions  of  a 
turn  are  read  on  the  divided  head.  As  contact  ap- 
proaches, the  screw  should  be  turned  with  a  very 
light  touch  and  the  same  force  used  for  different  contacts.  Some 
micrometers  are  provided  with  a  rachet  head  which  permits  only 
a  definite,  moderate  pressure. 

1 6.  Micrometer  Microscope. 

The  micrometer  microscope  is  a  microscope  with  cross-hairs 
at  the  focus.  In  one  type  of  instrument  these  cross-hairs  are 
movable  by  a  micrometer  screw.  In  the  other  and  more  com- 


14  MECHANICS. 

mon  type  the  whole  microscope  is  moved  by  a  micrometer  screw 
(see  Fig.  4).  The  most  elaborate  instruments  have  both  move- 
ments. The  rotations  of  the  screw  are  read  on  a  fixed  linear 
scale  while  the  fraction  of  a  rotation  is  read  by  a  circular  scale 
attached  to  the  screw,  and  thus  the  amount  of  movement  is 
ascertained  if  the  pitch  of  the  screw  is  known.  The  pitch  is  best 
determined  by  reading  a  length  on  a  reliable  scale  placed  in  the 
field  of  view. 

17.  Comparator. 

The  comparator  consists  essentially  of  a  pair  of  microscopes 
movable  along  a  horizontal  bar  to  which  they  are  at  right  angles. 
The  length  to  be  measured  is  placed  under  the  microscopes. 


FIG.  4. 

The  eye-piece  of  each  microscope  is  first  focused  clearly  on  the 
cross-hairs  and  the  whole  microscope  focused  without  parallax 
on  the  point  to  be  observed,  so  that  the  image  of  the  point  coin- 
cides with  the  intersection  of  the  cross-hairs.  The  object  is  then 
removed  and  a  good  scale  put  in  its  place,  and  a  reading  of  the 
scale  gives  the  required  length,  this  reading  being  facilitated  by 
the  use  of  micrometer  screws. 

1 8.  Spherometer. 

The  spherometer  is  an  instrument  with  four  legs,  three  of  which 
form  the  vertices  of  an  equilateral  triangle,  while  the  fourth  is 
at  the  center  of  the  triangle.  The  fourth  leg  can  be  screwed  up 
and  down  and  the  distance  of  its  extremity  from  the  plane  of  the 
extremities  of  the  other  three  legs  can  be  accurately  measured  by 


SPHEROMETER.  15 

means  of  a  linear  scale  attached  to  the  fixed  legs  and  a  circular 
scale  attached  to  the  movable  leg.  The  linear  scale  gives  the 
number  of  complete  turns  of  the  screw  and  the  circular  scale  the 
fraction  of  a  turn.  These  scales  are  read  when  the  screw  makes 
contact  with  an  object  placed  beneath. 

The  position  of  contact  may  be  determined  by  noticing  that  the 
screw  turns  very  easily  for  a  fraction  of  a  turn  just  after  contact 
begins.  This  is  due  to  reduced  friction  in  the  bearing,  owing  to 
the  weight  of  the  screw  falling  on  the  body  in  contact  and  to 
the  back-lash  of  the  screw,  the  frame  not  yet  being 
raised.  The  screw  should  be  lowered  until  it  thus 
begins  to  turn  very  easily ;  it  should  then  be  turned 
back  again  until  it  again  begins  to  turn  hard  and 
the  mean  reading  taken.  A  less  sensitive  method 
is  to  turn  the  screw  down  until  the  instrument  is 
felt  to  rock  or  wobble  and  make  a  reading;  then 
raise  the  screw  until  rocking  just  ceases,  make 
another  reading  and  take  the  mean  of  the  two  as  the 
contact  reading.  In  some  spherometers  the  end 
of  the  screw  on  making  contact  raises  two  levers 
arranged  to  greatly  magnify  the  motion.  The  screw  is  lowered 
until  the  top  lever  comes  to  some  definite  position,  for  instance, 
with  the  end  opposite  to  a  stud  in  the  frame  of  the  instrument. 

The  zero  reading  is  the  reading  of  the  scales  when  the  end  of  the 
screw  is  in  the  plane  of  the  ends  of  the  legs,  and  is  so  called  be- 
cause the  instrument  is  most  frequently  used  to  determine  dis- 
tances above  or  below  this  plane.  It  may  be  obtained  by  placing 
the  instrument  on  a  very  plane  plate  of  glass.  Several  zero 
readings  should  be  made  before  and  after  making  readings  with 
the  object  in  position  under  the  screw,  for  the  zero  reading  is 
likely  to  change  from  slight  disturbances  of  the  adjustment  of  the 
instrument  and  thermal  expansion  due  to  the  heat  of  the  hand. 
If  the  mean  of  the  zero  readings  made  after  reading  on  an  object 
should  be  decidedly  different  from  the  mean  of  those  made  before, 
readings  should  be  again  made  on  the  object  and  the  first  zero 
readings  discarded.  After  each  reading,  the  screw  should  be 
turned  up  through  at  least  a  quarter  revolution,  that  the  readings 
may  be  entirely  independent. 


1 6  MECHANICS. 

The  unit  of  the  linear  scale  may  be  obtained  by  comparison  with 
a  standard  steel  scale.  If  the  plane  of  the  circular  scale  is  not 
exactly  perpendicular  to  the  axis,  the  linear  scale  may  not  give 
the  correct  number  of  turns.  It  is  therefore  best,  as  a  check  upon 
the  linear  scale  readings,  to  count  the  number  of  rotations. 

19.  The  Dividing  Engine. 

The  dividing  engine  received  its  name  from  its  being  originally 
made  to  subdivide  scales,  diffraction-gratings,  etc.  An  equally 
frequent  use  of  the  instrument  is  for  the  accurate  measurement 
and  calibration  of  scales,  gratings,  etc. 

It  consists  essentially  of  (i)  a  very  carefully  made  horizontal 
screw,  the  ends  of  which  are  so  supported  that  the  screw  is  free 
to  rotate,  but  not  to  advance  or  recede;  (2)  a  nut  movable  on  the 
screw  and  bearing  against  (3)  a  platform  movable  along  a  track 
which  is  parallel  to  the  screw ;  (4)  a  micrometer  microscope  which 
in  some  instruments  is  held  in  a  support  movable  along  a  rail  on 
the  same  bed  plate  as  the  track,  but  in  other  instruments  is  car- 
ried on  the  platform;  (5)  dividing,  gear  for  making  scales,  etc.; 
(6)  a  divided  circular  scale  attached  to  the  screw  with  a  vernier 
attached  to  the  bed  plate. 

If,  by  means  of  the  circular  scale  and  vernier,  the  rotation  of 
the  screw  can  be  read  to  I  in  a  very  large  number,  say  to  I  in 
1,000,  then,  since  the  nut  moves  a  distance  equal  to  the  "pitch" 
of  the  screw  (measured  parallel  to  the  axis)  when  the  screw  is 
given  one  complete  rotation,  it  follows  that  the  movement  of  the 
nut  and  platform  can  be  read  to  a  correspondingly  small  fraction 
of  the  pitch  of  the  screw. 

The  object  whose  length  is  to  be  measured  is  placed  on  the 
movable  platform  (in  the  case  of  instruments  of  the  first  type 
mentioned  under  (4)  above).  The  microscope  is  focused  on 
one  end  of  the  length  to  be  measured,  so  that  the  intersection  of 
the  cross-hair  coincides  with  that  end.  By  turning  the  screw 
until  the  movement  of  the  platform  brings  the  other  end  of  the 
length  to  be  measured  into  coincidence  with  the  intersection  of 
the  cross-hairs  and  observing  the  number  of  turns  and  parts  of 
a  turn,  the  length  of  the  object  in  terms  of  the  pitch  of  the  screw 
as  unit  is  obtained.  The  true  pitch  of  the  screw  must  itself  be 


CATHETOMETER.  17 

obtained  by  comparing  it  by  the  same  method  with  some  ac- 
curately known  length,  such  as  a  length  on  a  standard  meter. 

The  adjustment  of  the  microscope  consists  of  two  steps:  (l) 
the  eye-piece  must  be  focused  on  the  cross-hairs  (but  the  eye- 
piece must  not  be  taken  out  lest  the  cross-hairs  be  injured);  (2) 
the  whole  microscope  muct  be  moved  toward  or  away  from  the 
object  until  it  is  seen  without  parallax,  i.  e.,  until  the  relative 
positions  of  the  cross-hairs  and  the  image  of  the  object  are  not 
changed  by  shifting  the  eye  sidewise.  The  length  to  be  measured 
must  then  be  placed  parallel  to  the  screw.  This  is  attained  when, 
by  rotation  of  the  screw,  the  image  of  each  end  can  be  brought 
to  coincidence  with  the  intersection  of  the  cross-hairs.  One  of 
the  most  frequent  sources  of  error,  in  using  a  measuring  instru- 
ment on  the  screw  principle,  is  back-lash  or  lost  motion.  To 
avoid  this  the  screw  should  always  be  turned  in  the  same  direc- 
tion during  a  measurement.  In  many  dividing  engines  back- 
lash is  impossible  because  the  motion  of  the  screw  cannot  be 
reversed,  the  platform  can  only  be  moved  in  the  reverse  direction 
by  unclasping  the  nut  (which,  for  this  purpose,  is  a  split  nut  held 
together  by  a  clasp  and  spring). 

Usually  the  handle  for  turning  the  screw  is  not  attached  to  the 
screw  or  circular  scale,  but  to  a  separate  disk  rotating  co-axially 
with  the  screw.  The  motion  of  the  handle  in  one  direction  is 
communicated  to  the  screw  by  a  ratchet;  when  the  handle  is 
reversed  the  ratchet  slips  freely.  As  an  aid  to  counting  the  num- 
ber of  turns  of  the  screw  in  measuring  a  considerable  length, 
two  detents  are  sometimes  geared  to  the  screw  in  such  a  way 
that  only  a  definite  number  of  turns  can  be  given  to  the  screw  at  a 
time,  after  which  the  handle  must  be  turned  back  for  beginning 
a  new  number  of  turns. 

A  more  complete  description  of  the  dividing  engine  will  be 
found  in  Stewart  and  Gee,  I,  §  16. 

20.  Cathetometer. 

The  cathetometer  is  a  vertical  pillar  supported  on  a  tripod  and 
leveling  screws,  and  capable  of  rotation  about  its  axis ;  the  pillar 
is  graduated  and  a  horizontal  telescope  with  cross-hairs  is  borne 
by  a  carriage  that  travels  on  the  pillar  and  can  be  clamped  at  any 


1 8  MECHANICS. 

desired  position.     A  slow-motion  screw  serves  for  accurate  ad- 
justment of  the  position  of  the  telescope. 

Adjustments.*  (i)  The  intersection  of  the  cross-hairs,  X,  must 
be  in  the  optical  axis  of  the  telescope.  To  secure  this,  focus  X  on 
some  mark,  rotate  the  telescope  about  its  own  axis  and  see 
whether  X  remains  on  the  mark.  If  not,  the  adjusting  screws 
of  the  cross-hairs  must  be  changed  until  this  is  attained. 

(2)  The  level  must  be  properly  adjusted.     Level  the  telescope 
until  the  bubble  comes  to  the  center  of  the  scale.     Turn  the  level 
end  for  end.     If  the  bubble  does  not  come  to  the  same  position, 
the  level  must  be  adjusted  until  it  will  stand  this  test. 

(3)  The  scale  must  be  vertical.     If  there  are  separate  levels  for 
the  shaft,  this  is  readily  attained.     If  there  is  but  one  level  for 
telescope  and  shaft,  this  and  the  next  adjustment  must  be  made 
simultaneously. 

(4)  The  telescope  must  be  perpendicular  to  the  scale.  The  top 
of  the  scale,  T,  may  be  regarded  as  having  two  degrees  of  freedom 
—first,  parallel  to  the  line  of  two  leveling  screws  af  the  base,  A 
and  B;  second,  in  a  line  through  the  third  leveling  screw,  C, 
perpendicular  to  AB.  If  A  and  B  be  screwed  equal  amounts  in 
opposite  directions,  T  will  move  parallel  to  AB.  If  C  only  be 
turned,  T  will  move  perpendicular  to  AB. 

First  make  the  telescope  horizontal  and  parallel  to  AB.  Turn 
the  shaft  through  180°.  It  is  easily  seen  that  if  the  telescope 
makes  an  angle  a  with  the  normal  to  the  scale,  turning  the  scale 
through  1 80°  will  cause  the  telescope  to  make  an  angle  2a  with 
its  former  direction.  Hence,  with  the  leveling  screw  of  the  tele- 
scope, correct  half  the  error  in  the  level,  and,  by  turning  A  and  B 
equally  in  opposite  directions,  correct  the  remainder.  Turn 
the  telescope  to  the  first  position  and  repeat  the  above  adjust- 
ments, then  to  the  second  and  continue  as  often  as  is  necessary. 
Then  turn  the  telescope  normal  to  AB  and  adjust  by  C.  When 
the  adjustment  is  complete,  turning  the  shaft  through  any  angle 
will  not  alter  the  position  of  the  bubble. 

Unless  the  cathetometer  is  on  a  perfectly  immovable  support, 
perfect  adjustment  is  not  possible  and  too  much  time  should  not 

*  Adjustments  (i)  and  (2)  are  not  usually  required  and  should  not  be  made 
without  the  advice  of  the  instructor. 


BAROMETER.  19 

be  spent  in  adjusting,  providing  the  telescope  is  accurately  level 
at  each  reading. 

The  eye-piece  of  the  telescope  is  focused  (but  not  removed) 
until  the  cross-hairs  seem  perfectly  distinct  and  the  focus  of  the 
objective  is  changed  until  the  object  is  seen  very  distinctly  and 
without  parallax,  i.  e.,  with  no  relative  motion  with  respect  to 
the  cross-hairs  when  the  eye  is  moved  about. 

21.  Barometer. 

References — Elementary:  Duff,  §§219-220;  Ames,  pp.  176-177;  Crew,  §163; 
Kimball,  §§189,  191;  Reed  &  Guthe,  §71;  Spinney,  §§116-117;  Watson, 
§§132-135 — More  adianced:  McLean's  Practical  Physics,  §§98-101;  Wat- 
sons Practical  Physics,  §58. 

If  the  barometer  is  of  Fortiris  cistern  form,  the  cistern  is  raised 
or  lowered  by  means  of  the  screw  at  the  bottom  until  the  mercury 
just  meets  an  ivory  stud  near  the  side  of  the  cistern.  A  collar 
to  which  is  attached  a  vernier  is  adjusted  until  the  top  of  the 
meniscus  of  the  mercury  column  is  tangent  to  the  plane  of  the 
two  lower  edges.  The  height  of  the  barometer  should  be  reduced 
to  zero  by  the  formula 

h0  =  h(i  —  .0001621) 

where  h  is  the  observed  height,  ho  the  height  at  o°  and  t  the 
temperature  Centigrade.  For  the  expansion  of  the  mercury 
will  increase  the  height  in  the  ratio  (i.  +  .0001 8 it)  and  the  ex- 
pansion of  the  brass  scale  will  reduce  the  apparent  height  in  the 
ratio  (i  -  .0000190  (Table  VII). 

The  siphon  barometer  has  two  scales,  graduated  on  the  glass 
tube,  in  opposite  directions,  from  a  common  zero.  The  length 
of  the  mercury  column  is  obviously  the  sum  of  the  readings  of 
the  mercury  levels  in  the  two  tubes.  Since  the  coefficient  of 
linear  expansion  of  glass  is  only  about  0.000008,  the  correction 
formula  becomes 

fc0  =  &(i -.0001 730 . 

Since  the  mercury  may  adhere  to  the  glass  to  some  extent, 
barometer  tubes  should  be  tapped  gently  before  reading. 


2O  MECHANICS. 

22.  The  Balance. 

Kohlrausch,  §§7-11;   McLean's  Practical  Physics,  §§26-37;   Watson's  Practical 

Physics,  §§  25,  26. 

Weighing  by  a  Sensitive  Balance. — On  first  using  a  sensitive 
balance  note  the  position,  purpose,  and  structure  of  the  following 
parts : — 

The  beam,  The  knife-edges  and  planes, 

The  pointer,  The  arrestment, 

The  pillar,  The  rider-arms, 

The  pan-supports. 

By  the  sensibility  of  a  balance  is  meant  the  amount  of  deflection 
of  the  beam  produced  by  a  given  small  weight.  Consider  how  the 
sensibility  depends  on  (i)  the  length  of  the  beam,  (2)  the  weight 
of  the  beam  and  pans,  (3)  the  distance  of  the  point  of  suspension 
of  the  beam  from  its  center  of  gravity.  Will  the  sensibility  of  a 
certain  balance  be  different  with  different  loads  on  the  pans  and 
why?  How  can  the  sensibility  be  varied  with  a  given  load? 
(See  references.) 
Precautions  in  Use  of  Balance. 

1 .  Note  the  maximum  load  that  may  be  placed  on  the  balance 
and  take  care  not  to  exceed  it. 

2.  Always  lift  the  beam  from  the  knife-edges  before  in  any  way 
altering  the  load  on  the  pans. 

3.  Do  not  stop  the  swinging  of  the  balance  with  a  jerk.     It  is 
best  to  stop  it  when  the  pointer  is  vertical. 

4.  To  set  the  beam  in  vibration,  do  not  touch  it  with  the  hand, 
but  raise  and  lower  the  arrestment. 

5.  Place  the  large  weights  in  the  center  of  the  pan. 

6.  Make  final  weighings  with  the  case  closed. 

7.  Replace  all  weights  in  their  proper  place  in  the  box  when  they 
are  not  actually  in  use.     Do  not  use  weights  with  different  boxes. 

8.  Do  not  place  anything  in  contact  with  a  pan  that  is  liable 
to  injure  it. 

9.  Avoid,  if  possible,  weighing  a  hot  body. 

10.  Never  handle  the  weights  with  the  fingers,  as  this  may 
change  some  of  the  weights  appreciably.     Always  use  the  pincers. 

Notice  the  dimensions  of  the  weights  in  the  box,  e.  g.,  50  g., 
20  g.,  10  g.,  10  g.,  5  g.,  2  g.,  i  g.,  i  g.,  etc.  Instead  of  weights 


THE    BALANCE.  21 

of  0.005  g.,  0.002  g.,  o.ooi  g.,  o.ooi  g.,  it  is  customary  to  use  a 
rider  of  o.oi  g.,  which  can  be  placed  on  the  beam  at  various  dis- 
tances from  the  center.  The  beam  is  for  this  purpose  graduated 
into  10  divisions,  which  may  be  still  further  subdivided.  Thus 
the  o.oio  g.  rider  placed  at  the  division  4  of  the  beam  is  equivalent 
to  0.004  g.  placed  on  the  pan. 

The  zero-point  of  the  balance  is  the  position  on  the  scale  behind 
the  pointer  at  which,  the  pans  being  empty,  the  pointer  would 
ultimately  come  to  rest;  it  must  not  be  confused  with  the  zero 
of  the  scale.  As  much  time  would  be  wasted  in  always  waiting 
for  the  pointer  to  come  to  rest,  the  zero  of  the  balance  is  best 
obtained  from  the  swings  of  the  pointer.  For  this  purpose, 
readings  of  the  successive  "turning-points"  are  made  as  follows — 
three  successive  turning-points  on  the  right  and  the  two  inter- 
mediate ones  on  the  left,  or  vice  versa;  e.  g., 

Turning  points. 


L. 

R. 

-1-3 

+  2.1 

-  i.i 

+  2.0 

-  i.o 

Mean, 

-  1.13 

+  2.05 

-I.I3 

Zero-point 

= 

+  0.92   -v-  2   =    +  0.46. 

By  taking  an  odd  number  of  successive  turning-points  on  one 
side  and  the  intermediate  even  number  on  the  other  side  and  then 
averaging  each  set,,  we  eliminate  the  effect  of  the  gradual  decrease 
of  amplitude  of  the  swing. 

The  resting-point  of  the  balance  with  any  loads  on  the  pans  is 
the  point  at  which  the  pointer  would  ultimately  come  to  rest, 
and  is  found  in  the  same  way  as  the  zero-point.  If  the  resting- 
point  should  happen  to  be  the  same  as  the  zero-point,  the  weight 
of  the  body  on  one  pan  is  immediately  i 
found  by  the  weights  on  the  other  pan  and  " 
the  position  of  the  rider.  Usually,  how- 
ever, this  will  not  be  so.  With  the  rider  at 
a  suitable  division,  find  the  resting-point  on  one  side  of  the  zero- 


22  MECHANICS. 

point,  and  then,  after  altering  the  rider  one  place,  find  the  resting- 
point  on  the  other  side  of  the  zero.  By  interpolation  the  change 
of  the  position  of  the  rider  necessary  to  make  the  resting-point 
coincide  with  the  zero-point  is  deduced.  For  example,  the 
zero  is  +0.46;  with  the  rider  at  4  the  resting-point  is  +0.51; 
with  the  rider  at  5  the  resting-point  is  +0.10.  By  changing  the 
rider  from  4  to  5,  o.ooi  g.  was  added.  To  bring  the  resting-point 
to  the  zero  we  should  have  added  0.05  -f-  (0.510  —  .10)  of  o.ooi  g. 
or  o.oooi  g.  approximately.  Hence  the  weight  of  the  body  is  the 
weight  on  the  pan  plus  0.0041  g. 

The  arms  of  the  balance  may  be  unequal.  If  this  be  so,  the 
weight  obtained  above  will  not  be  the  true  weight.  To  eliminate 
this  error  the  body  must  be  changed  to  the  other  pan  and  another 
weighing  made.  If  /  be  the  length  of  the  left  arm  and  r  that  of 
the  right  and  if  u  be  the  counterbalancing  weight  when  the  body 
is  in  the  left  pan  and  v  when  it  is  in  the  right,  while  w  is  the  true 
weight  of  the  body  then, 


(The  geometric  mean  of  two  very  nearly  equal  quantities  is 
nearly  equal  to  their  arithmetic  mean.)  The  ratio  of  the  arms 
of  the  balance  may  also  be  calculated,  since 


« 


The  buoyancy  of  the  air  on  the  weights  and  on  the  body  must 
be  allowed  for  in  accurate  work.  To  the  apparent  weight  of  the 
body  must  be  added  a  correction  equal  to  the  weight  of  the  air 
displaced  by  the  body  and  from  the  apparent  weight  must  be 
subtracted  the  weight  of  the  air  displaced  by  the  weights.  In 
each  case  the  weight  of  the  air  displaced  can  be  calculated  if  its 
volume  and  density  are  known.  This  correction  in  any  case  is 
very  small.  A  small  percentage  error  in  the  correction  will  not 
appreciably  affect  the  calculated  true  weight.  Hence  approxi- 
mate values  of  the  volumes  of  the  body  and  weights  may  be  used. 
In  finding  the  volume  of  the  weights  the  density  of  brass  weights 
may  be  taken  as  8.4.  The  density  of  air  at  o°  and  760  mm.  may 


ADJUSTMENT  OF  TELESCOPE  AND  SCALE.          23 

be  taken  as  .0013,  and  its  density  at  the  temperature  of  the  labora- 
tory and  the  pressure  indicated  by  the  barometer  may  be  calcu- 
lated by  the  laws  of  gases.  Hence  the  temperature  and  baro- 
metric pressure  should  be  obtained. 

23.  Adjustment  of  Telescope  and  Scale. 

To  adjust  a  telescope  and  scale,  determine  approximately  the 
location  of  the  normal  to  the  mirror,  either  by  finding  the  image 
of  one  eye  or  the  image  of  an  incandescent  lamp  held  near  the  eye. 
Move  the  stand  supporting  the  telescope  and  scale  until  the  center 
of  the  scale  is  about  in  line  with  the  normal.  Look  along  the 
outside  of  the  telescope  at  the  mirror  and  move  the  scale  up  and 
down,  or,  if  this  is  not  possible,  raise  or  lower  the  stand  until  you 
see  the  reflection  of  the  scale  in  the  mirror.  It  may  be  a  help  to 
illuminate  the  scale  with  an  incandescent  lamp.  Focus  the  tele- 
scope on  the  mirror,  then  change  the  focus  until  the  scale  is  seen, 
remembering  that  the  image  of  the  scale  is  at  about  twice  the 
distance  of  the  mirror  and  that  the  more  distant  the  object  the 
more  must  the  eye-piece  be  pushed  in.  Having  found  the  scale 
adjust  the  eye-piece  until  the  cross-hairs  are  very  clear  and  then 
focus  the  telescope  until  there  is  no  parallax  between  the  cross- 
hairs and  the  image. 

24.  Time  Signals. 

A  convenient  source  of  time  signals  for  a  laboratory  is  a  chrono- 
meter which  either  opens  or  closes  a  circuit  containing  batteries, 
sounders,  etc.,  every  second  with  an  omission  at  the  end  of  each 
minute. 

The  individual  second  intervals  indicated  by  a  chronometer, 
so  arranged,  are  likely  to  be  somewhat  inaccurate,  and  therefore, 
when  an  accurate  interval  of  one  second  is  required,  a  second's 
pendulum  should  be  used  with  a  platinum  point  making  contact 
with  a  drop  of  mercury,  and  thus,  if  desired,  closing  an  electric 
circuit.  Since  it  is  difficult  to  set  the  mercury  drop  exactly  in 
the  center  of  the  path,  alternate  seconds  are  likely  to  be  too  long. 
Therefore,  if  possible,  a  two  seconds'  interval  should  be  employed, 
alternate  contacts  being  disregarded.  If  these  contacts  cause 
confusion,  a  pendulum  omitting  alternate  contacts  may  be  used 
(see  Ames  and  Bliss,  p.  486). 


24  MECHANICS. 

I.   TO  MAKE  AND  CALIBRATE  A  SCALE. 

To  illustrate  the  use  of  the  dividing  engine  (described  on  page 
1 6)  a  short  scale  is  to  be  engraved  in  millimeters  on  a  strip  of 
nickel-plated  steel  and  then  calibrated  by  comparison  with  the 
average  millimeter  of  a  standard  scale. 

Arrange  the  cogs  of  the  dividing  gear  so  that  each  fifth  mm. 
division  shall  be  longer  than  the  intermediate  divisions  and  each 
tenth  division  still  longer.  Test  this  adjustment  on  a  rough  test 
strip.  Next  clamp  the  strip  to  be  divided  on  the  platform  of 
the  engine  so  that  it  is  parallel  to  the  screw;  this  can  be  tested 
by  observing  the  edge  of  the  strip  in  the  microscope  as  the  plat- 
form is  advanced  by  the  screw.  Care  should  be  taken  to  clamp 
the  pillar  that  supports  the  divider  so  that  the  point  of  the  divider 
moves  perpendicular  to  the  length  of  the  scale.  A  scale  of  2  or 
3  cms.  should  then  be  marked  out  on  the  steel  strip  and  the 
temperature  of  the  platform  ascertained  by  a  thermometer. 

This  scale  is  next  to  be  calibrated.  The  exact  pitch  of  the 
screw  is  first  obtained  in  terms  of  the  mm.  of  the  standard.  For 
this  purpose  a  considerable  length,  e.  g.,  a  decimeter,  of  the  stand- 
ard should  be  measured  on  the  engine.  This  should  be  done  for 
three  different  parts  of  the  screw.  The  agreement  of  the  three 
determinations  will  afford  some  indication  of  the  uniformity  of 
the  screw.  The  scale  should  then  be  measured  mm.  by  mm.  For 
the  first  reading,  the  circular  scale  of  the  screw  may  be  set  to 
zero  when  the  cross-hairs  coincide  with  the  zero  division  of  the 
scale  to  be  measured,  and  thereafter  the  screw  should  be  turned 
always  in  the  same  direction  and  only  arrested  for  a  reading  of  the 
circular  scale  and  vernier  (the  total  number  qf  turns  being  also 
noted)  when  the  microscope  shows  that  the  middle  of  a  division 
has  come  to  coincide  with  the  intersection  of  the  cross-hairs. 
As  this  coincidence  approaches,  the  handle  should  be  turned 
slowly,  and  if  turned  too  far  the  reading  at  that  point  must  be 
omitted  altogether.  The  handle  should  also  be  turned  slowly 
when  contact  with  the  detent  approaches  so  that  the  screw  may 
not  be  arrested  with  a  jerk. 

As  a  check  on  the  work,  the  whole  length  of  the  scale  should  be 
measured. 


ERRORS    OF   WEIGHTS.  25 

In  calculating  the  true  length  of  the  divisions,  allowance  must 
be  made  for  the  temperature  of  the  standard  which  may  be  taken 
as  the  temperature  of  the  platform  of  the  engine.  The  standard 
is  correct  at  the  temperature  marked.  From  its  coefficient  of 
expansion  calculate  the  length  of  its  mm.  at  the  temperature  of 
observation  and  then  deduce  the  pitch  of  the  screw  at  the  same 
temperature.  Then  from  the  readings  made,  calculate  the  length 
of  each  millimeter  of  the  scale  and,  by  addition,  draw  up  a  table 
showing  the  true  distance  of  each  division  from  the  zero  division. 

Questions. 

1.  Enumerate  the  possible  sources  of  error  in  the  use  of  the  dividing  engine 
for  the  manufacture  of  scales. 

2.  At  what  temperature  would  the  whole  length  of  your  scale  be  an  exact 
number  of  centimeters?     (Table  VII.) 


II.    ERRORS  OF  WEIGHTS. 

Kohlrausch,§i2;  McLean's  Practical  Physics,  §37;  Watson's  Practical  Physics, 

§27- 

Weights  by  good  makers  are  usually  so  accurate  that  errors 
in  them  may  for  most  purposes  be  neglected.  But  when  less 
perfect  weights  are  to  be  used  or  when  weighings  are  to  be  made 
with  the  highest  possible  degree  of  accuracy,  the  errors  in  the 
weights  must  be  carefully  ascertained. 

We  shall  suppose  that  a  100  g.  box  of  weights  is  to  be  tested, 
and  that  a  reliable  100  g.  weight  is  supplied  as  a  standard,  and 
that  an  accurate  10  mg.  rider  is  supplied  for  making  the  weighings. 
The  weights  of  the  box  will  be  denoted  by  100',  50',  20',  20",  10', 
and  so  on,  and  the  sum  5'  +  2'  +  2"  +  i'  by  10".  To  find  the 
six  unknown  quantities,  100',  50',  20',  20",  10',  10",  we  must 
make  six  weighings  and  obtain  six  relations  between  these  quanti- 
ties. Such  a  set  of  weighings  are  indicated  in  the  following  table, 
Each  should  be  performed  by  the  method  of  double-weighing 
described  on  page  22. 

10'    =  10"  +  a 

20'    =  10'    +  10"  +  b 

20"  =  20'    +  c 

50'    =  20'    +  20"  +  10'   +  d 
100'    =  50'    +  20'   +  20"  -f  10'  +  e 
100     =  100'  -J-/ 


26  MECHANICS. 

To  solve  these  equations,  substitute  the  value  of  10'  given  by 
the  first  in  the  second ;  then  substitute  the  value  of  20'  given  by 
the  second  in  the  third,  and  so  on  to  the  last,  when  the  value  of 
10"  in  terms  of  the  standard  100  and  a,  b,  c,  d,  e,  f  will  be  obtained. 
The  calculation  of  the  other  quantities  will  then  present  no  diffi- 
culty. To  standardize  the  box  completely  the  same  process 
must  be  applied  to  10',  5',  2',  2",  i',  i",  and  similarly  to  the 
smaller  weights. 

III.  VOLUME,  MASS,  AND  DENSITY  OF  A  REGULAR 

SOLID. 

The  mass  of  the  specimen  (a  sphere  or  cylinder)  is  found  by 
weighing  on  a  sensitive  balance  (see  p.  20).  To  eliminate  the 
inequality  of  the  arms  of  the  balance,  the  body  should  be  weighed 
in  both  pans  (p.  22).  The  zero-point  and  resting-points  of  the 
balance  should  be  found  by  the  method  of  vibrations  and  the 
various  precautions  in  the  use  of  the  balance  must  be  carefully 
observed.  Allowance  should  be  made  for  air  buoyancy  (p.  22) 
and  for  this  purpose  the  barometer  (p.  19)  and  temperature 
should  be  observed.  Corrections  should  also  be  applied  to  the 
weights,  if  the  weights  have  been  corrected  in  the  preceding 
experiment,  or  if  a  table  of  corrections  is  supplied. 

The  dimensions  of  the  specimen  are  measured  by  a  micrometer 
caliper  (p.  13)  or  a  vernier  caliper  (p.  13).  If  the  body  is  spheri- 
cal, ten  measurements  of  the  diameter  should  be  made  and  the 
average  taken;  if  it  is  cylindrical  ten  measurements  of  the  di- 
ameter and  ten  of  the  length  should  be  made. 

From  the  mass  and  the  volume,  the  density  (or  mass  per  c.c.) 
is  deduced. 

The  ratio  of  the  arms  of  the  balance  should  also  be  derived 
from  the  results  of  the  double  weighing  (p.  22). 

The  possible  error  of  the  density  determination  should  be 
calculated  as  illustrated  on  p.  6. 

Questions. 

1.  If  the  object  aimed  at  were  merely  the  density  of  the  body,  which  of  the 
above  measurements  should  be  improved  in  precision  and  to  what  extent  would 
it  need  to  be  improved? 

2.  If  the  above  improvement  were  not  possible,  how  much  of  the  refinement 
of  measurement  of  the  other  quantity  might  be  discarded? 

3.  What  density  must  a  body  have  to  make  the  correction  in  air  bouyancy 
negligible? 


MOHR-WESTPHAL    SPECIFIC   GRAVITY    BALANCE. 


IV.   MOHR-WESTPHAL  SPECIFIC  GRAVITY  BALANCE. 

Kohlrausch,  p.  45;  McLean's  Practical  Physics,  §62;  Stewart  and  Gee,  I,  §92,  III. 

This  is  a  convenient  form  of  hydrostatic  balance  for  finding 
the  density  of  a  liquid  by  determining  the  buoyancy  of  the  liquid 
on  a  float  hung  from  an  arm  of  the  balance  and  immersed  in  the 
liquid.  Instead  of  weights  riders  are  used,  the  arm  of  the  balance 
from  which  the  float  hangs  being  graduated  into  ten  divisions. 
The  float  is  made  of  such  a  size  that  when  hanging  in  air  from  the 
graduated  arm  of  the  balance  (which  is  less  massive  than  the 
other  arm)  it  will  just  produce  equilibrium.  Four  riders  of  differ- 
ent mass  are  employed,  each  one  being  ten  times  as  heavy  as 
the  next  smaller.  The  largest  rider  is  of  such  a  size  that  if  the 
float  hanging  from  the  bal- 
ance be  immersed  in  water 
at  15°  C.  the  addition  of 
the  rider  to  the  hook  at 
the  end  of  the  beam  will 
restore  equilibrium.  Hence 
it  counter-balances  the 
buoyancy  of  the  water  on 
the  float.  Thus  it  is  evi- 
dent that  if  the  water  be 
replaced  by  a  liquid  of  un- 
known density  at  the  same 
temperature  (so  that  the 
volume  of  the  float  is  the  same)  and  if  the  largest  rider  under 
the  circumstances  produces  equilibrium  when  placed  at  the  sixth 
division,  then  for  equal  volumes,  this  liquid  can  weigh  only 
six-tenths  as  much  as  water,  or  its  density  is  0.6.  A  second  rider, 
one-tenth  as  heavy  as  the  first,  would  evidently  enable  us  to  carry 
the  process  one  decimal  place  farther,  etc.  For  liquids  of  a 
density  exceeding  unity,  another  rider  equal  to  the  largest  must 
be  hung  from  the  end  of  the  beam,  and  still  a  third  may  be 
necessary  for  liquids  of  density  above  2. 

From  the  above  it  will  be  seen  that  (i)  the  balance  must  be 
adjusted  by  the  leveling  screw  on  the  base  until  the  end  of  the 
beam  is  opposite  the  stud  in  the  framework  when  the  float  is 
suspended  in  the  air;  (2)  the  beaker  must  always  be  filled  to  the 


FIG.  7. 


28  MECHANICS. 

same  level,  that  level  being  such  that  when  the  liquid  is  water 
at  15°  C.  the  balance  is  in  equilibrium  with  the  largest  rider 
hanging  above  the  float,  and  (3)  the  liquid  tested  must  be  at 

15°  c. 

As  an  exercise  in  the  use  of  this  balance,  find  what  shrinkage 
of  volume  there  is  in  the  solution  of  some  salt  (e.  g.,  common  salt, 
ammonium  chloride  or  copper  sulphate)  in  water  and  find  how 
the  shrinkage  varies  with  the  concentration.  Solutions  may  be 
made  up  by  weighing  out  very  carefully  on  a  sensitive  balance 
(see  p.  20),  0.5  gm.,  I  gm.,  4  gm.,  10  gm.,  etc.,  of  the  salt  and  dis- 
solving each  in  a  deciliter  of  water.  When  the  density  of  a  solu- 
tion has  been  found,  the  percentage  contraction  is  calculated 
from  the  sum  of  the  volumes  of  the  constituents  before  mixture 
and  the  volume  of  the  solution  after  mixture;  the  volume  in  each 
case  equals  the  mass  divided  by  the  density.  The  densities  of 
various  salts  are  given  in  Table  VII. 

The  densities  found  and  the  percentages  of  contraction  should 
be  represented  by  curves  with  percentages  of  salt  as  abscissae. 
If  any  determination  of  density  be  largely  in  error  it  will  be  shown 
by  the  curve. 

If  time  permit,  determine  the  density  at  15°  of  equivalent 
solutions*  of  several  salts  having  the  same  base,  e.  g.,  NaCl ; 
1/2  Na2SO4  ;  NaNO3  ;  etc.,  and  compare  with  the  densities  of 
similar  solutions  with  a  different  base,  e.  g.,  NH4C1;  1/2  (NH4)2 
SO4,  NH4NO3,  etc.  The  difference  in  density  between  cor- 
responding salts  should  be  approximately  constant  (Valsons 
Law  of  Moduli).  Find  similarly  the  difference  in  densities  con- 
tributed by  the  acid  radicals,  e.  g.,  NaNO3  and  NaCl;  NH4NO3 
and  NH4C1,  etc. 

Questions. 

1.  What  sources  of  error  may  there  be  in  a  determination  of  density  by  this 
method? 

2.  How  might  the  accuracy  of  the  riders  be  tested? 

3.  How  might  the  accuracy  of  graduation  of  the  beam  be  tested? 

4.  What  effect  has  capillarity? 

5.  Explain  the  Law  of  Moduli,  f 

6.  What  were  the  greater  and  the  least  densities  which  could  be  measured? 

*  The  chemical  equivalent  of  a  substance  is  the  atomic  or  melocular  weight 
divided  by  the  valency.  Two  solutions  are  equivalent  if  the  number  of  grams 
of  each  dissolved  in  one  liter  (or  that  proportion)  is  the  same  fraction  of  the 
respective  chemical  equivalent. 

t  Phy.  Chem.,  Ewell,  p.  159. 


DENSITY   BY   VOLUMENOMETER.  29 


V.    DENSITY  BY  VOLUMENOMETER. 

References — Elementary:  Duff,  §221 ;  Crew,  §170;  Kimball,  §199;  Reed  and  Guthe, 
§77;  Spinney,  §122;  Watson,  §130. — More  Advanced:  Gray's  Treatise  on 
Physics,  I,  §426. 

When  the  density  of  such  substances  as  gunpowder,  sugar, 
starch,  etc.,  is  to  be  determined,  neither  the  method  of  immer- 
sion in  a  liquid  nor  that  of  the  direct  measurement  of  mass  and 
volume  can  be  employed.  The  method  then  usually  employed 
is  that  of  the  volumenometer.  This  is  a  method  of  immersion  in 
air  instead  of  immersion  in  water,  with 
an  application  of  Boyle's  Law  instead 
of  Archimedes'  principle.  The  volume 
of  the  body  is  found  by  placing  it  in  a 
glass  vessel  and  noting  how  much  the 
pressure  in  the  vessel  changes  when 
the  air  is  allowed  to  expand. 

A  thick  walled  test  tube  of  about  30  . 

c.c.  capacity,  A,  into  which  the  sub-    C  v 

stance  may  be  introduced,  is  connected 
with  an  open  mercury  manometer  (see 
Fig.  8) .  The  rubber  stopper  should  be 
forced  into  the  test  tube  to  a  definite 
mark.  If  leakage  is  found  the  joints 
may  be  made  tight  with  rubber  grease.* 
DE  is  raised  until  the  mercury  in  the 
burette  BK  is  at  a  division  B  which 
is  carefully  observed.  The  pressure, 
P,  in  A  is  carefully  detei  mined  from 
the  difference  in  mercury  levels  and  _ 

the  barometer.     By  the  use  of  a  rear 

mirror,  parallax  may  be  avoided  and  a  small  square  will  assist 
in  reading  a  scale  between  the  two  arms  of  the  manometer. 
The  accuracy  of  the  readings  may  be  increased  by  using  a 
cathetometer  (p.  17).  Lower  DE  until  the  mercury  is  at  a  divi- 

*  Equal  parts  pure  rubber  gum,  vaseline,  and  paraffin.  The  two  latter  are 
melted  together  and  the  rubber  is  cut  into  small  pieces  and  dissolved  in  the 
heated  liquid. 


3O  MECHANICS. 

sion  K  and  again  determine  the  pressure,  p.  The  particular 
division  chosen  should  be  such  that  the  pressure  p  is  approximately 
one-half  of  P.  Let  the  volume  between  B  and  K  be  v.  Let  V  be 
the  volume  of  A,  and  connecting  tubing,  to  B.  By  Boyle's  Law: 
PV  =  p(V  +  v) 

Make  at  least  six  determinations  of  P  and  p,  bringing  the 
mercury  each  time  to  the  same  points  B  and  K.  Calculate  V 
from  the  mean  values. 

Now  introduce  a  carefully  weighed  amount  of  the  assigned 
powder  into  the  test  tube  A  and  insert  the  stopper  b  to  its 
former  depth.  Again  determine  the  pressure  P'  when  the 
mercury  level  is  at  B  and  observe  the  pressure  p'  when  the  volume 
is  increased  an  amount  v'  such  that  p'  is  about  one-half  of  P' . 
Repeat  as  before.  If  x  is  the  unknown  volume  of  the  powder, 
the  previous  equation  becomes 

P'  (V-x)  =  p'  (V-x  +  z/) 

from  which  x  may  be  calculated.     From  the  volume  and  mass  of 
the  powder  its  density  is  determined. 

If  time  permit,  determine  the  density  also  with  a  specific 
gravity  flask  (pyknometer).  Weighings  should  be  made  of  (i) 
bottle  empty;  (2)  bottle  filled  with  a  liquid  of  known  density 
which  is  inert  toward  the  body,  and  (3)  with  a  known  mass  of  the 
body  in  it,  the  rest  of  the  bottle  being  filled  with  the  liquid.  An 
equation  for  density  can  be  worked  out. 

The  possible  error  in  the  determination  of  the  density  is  found 
by  methods  explained  on  pages  4-9. 

Questions. 

1.  What  sources  of  error  remain  uneliminated? 

2.  Prove  that  for  the  best  results  p  should  be  one-half  of  P. 

VI.    DENSITY  OF  AIR. 

References — Elementary:  Crew,  §170;  Duff,  §221;  Kimball,  §199;  Reed  &  Guthe, 
§77;  Spinney,  §122;  Watson,  §130. — More  Advanced:  McLean's  Practical 
Physics,  §72. 

The  density  of  air  at  atmospheric  pressure,  or  its  mass  per  cubic 
centimeter,  might  be  obtained  by  weighing  a  flask  containing  air 
at  atmospheric  pressure  and  then  re-weighing  it  after  all  the  air 


DENSITY   OF   AIR.  31 

has  been  removed  by  an  air-pump.  The  difference  of  weight, 
together  with  the  volume  of  the  flask,  would  give  the  density  of 
the  air.  In  practice  the  procedure  has  to  be  modified,  because 
it  is  impossible  to  completely  exhaust  the  flask  of  air.  The 
modification  consists  in  finding  the  pressure  of  the  air  remaining 
in  the  flask  and  taking  account  of  it. 

Let  D  be  the  required  density  at  the  room  temperature  and 
pressure,  P.  Let  d  be  the  density  of  the  remaining  air  when  the 
pressure  has  been  reduced  to  p.  Let  the  weight  of  the  flask 
when  filled  with  air  be  W  and  let  w  represent  its  weight  when 
exhausted  to  the  pressure  p. 

W-w  =  V  (D-d) 
By  Boyle's  Law 

~D  =  P 

Hence 

D=W-w      P 


V     'P-p' 

A  convenient  form  of  flask  is  a  round-bottom  flask  from 
which  part  of  the  neck  has  been  cut  off  and  which  is  closed  by  a 
rubber  stopper  containing  a  glass  tube  with  a  glass  stop-cock. 
The  rubber  stopper  will  hold  tighter  if  lubricated  with  rubber 
grease*  before  insertion. 

If  the  flask,  as  found,  is  dry,  it  will  be  better  to  postpone  finding 
its  volume  until  the  end  of  the  experiment,  as  the  operation  re- 
quires it  to  be  filled  with  water.  Moreover,  of  the  two  weighings 
for  finding  the  mass  of  air  removed,  it  is  better  to  make  the  one 
with  the  flask  partly  exhausted  first,  for  the  weighing  with  the 
air  admitted  can  be  made  immediately  after,  without  handling 
the  flask  or  removing  it  from  the  balance,  a  point  of  some  im- 
portance where  the  difference  of  weight  to  be  measured  is  so 
small.  To  save  delay  in  weighing  the  flask  after  it  has  been 
exhausted,  the  zero  reading  of  the  fine  balance  used  should  be 
obtained  before  the  flask  is  exhausted.  For  the  method  of 
accurate  weighing,  by  oscillations,  see  page  21. 

*  See  note,  p.  28. 


32  MECHANICS. 

A  Bunsen's  aspirator  or  a  Geryk  pump  is  satisfactory  for  ex- 
hausting the  flask.  The  flask  should  be  connected  to  the  as- 
pirator or  pump,  through  a  bottle  for  catching  any  water  or 
mercury.  An  open-tube  manometer  connected  to  the  tube  that 
joins  the  aspirator  or  pump  and  flask  will  give  the  pressure. 

There  should  be  a  stop-cock  or  a  rubber  pinch-cock  in  the  con- 
nection between  the  manometer  and  the  pump  or  aspirator. 
When  a  sufficiently  high  exhaustion  has  been  secured  this  cock 
should  be  closed  for  several  minutes  to  ascertain  if  there  is  any 
leakage.  If  not,  both  ends  of  the  manometer  should  be  read  and 
the  stop-cock  of  the  flask  closed.  Before  removal  of  the  flask, 
the  other  cock  should  be  opened  that  the  rest  of  the  apparatus 
may  fill  with  air.  If  by  any  chance  a  small  quantity  of  water 
should  pass  into  the  manometer  from  the  aspirator,  allowance 
should  be  made  for  it,  the  density  of  mercury  being  taken  as  13.6. 

The  flask  is  then  weighed  as  quickly  as  possible  on  a  fine 
balance,  the  method  of  vibration  being  used.  It  may  be  neces- 
sary to  hang  the  flask  by  a  fine  wire  to  the  hook  which  carries 
the  pan.  This  weighing  is  repeated  with  the  stop-cock  open, 
but  with  the  flask  otherwise  undisturbed.  The  atmospheric 
pressure  is  obtained  from  the  barometer  (see  p.  19),  and  the 
temperature  from  a  thermometer  hung  inside  the  balance  case. 

The  volume  of  the  flask  may  be  obtained  by  filling  it  with 
distilled  water  and  weighing  it  on  an  open  balance.  To  get  the 
flask  just  filled  to  the  stop-cock,  the  stopper  (removed  for  filling 
the  flask)  should  be  thrust  in  with  the  stop-cock ;  open  the  stop- 
cock should  then  be  closed,  and  any  water  above  the  stop-cock 
should  be  removed.  Of  course,  the  stop-cock  should  be  replaced 
at  its  original  depth,  which  should  be  marked.  The  density  of 
water  at  different  temperatures  will  be  found  in  Table  V. 

When  the  experiment  is  completed,  place  the  open  flask  in- 
verted on  a  frame  to  dry,  so  that  it  may  be  ready  for  the  next 
person  who  uses  it. 

The  density  of  dry  air  may  be  found  in  the  same  way,  the  flask 
being  several  times  exhausted  and  refilled  through  a  drying-tube. 
Similarly  the  density  of  any  other  gas,  e.  g.,  carbon  dioxide,  may 
be  found  by  filling  the  flask  from  a  generator.  The  gas  must  be 
admitted  to  the  exhausted  flask  very  slowly  and^the  exhaustion 


ACCELERATION   OF   GRAVITY   BY   PENDULUM.  33 

and  filling  must  be  repeated  to  insure  the  (almost)  complete  re- 
moval of  the  air. 

In  reporting,  deduce  from  your  measurement  of  the  density 
of  air  or  gas  its  density  at  o°  C.  and  760  mm.  by  using  Boyle's 
and  Charles'  Laws.  Find  also  the  possible  error  of  the  measure- 
ment of  density  (p.  5). 

Questions. 

1.  Would  the  first  results  be  affected  by  the  presence  of  water  in  the  flask? 
Explain. 

2.  Should  the  flask  weigh  more  filled  with  dry  air  or  filled  with  moist  air, 
both  at  atmospheric  pressure?     Why? 

3.  What  error  would  be  caused  by  an  uncertainty  of  2  mm.  in  the  position 
of  the  rubber  stopper? 

4.  What  other  methods  are  there  for  obtaining  the  density  of  air? 

VII.    ACCELERATION   OF   GRAVITY  BY  PENDULUM. 

References — Elementary:  Duff,  §§117,  120;  Ames,  pp.  74,  91,  135;  Crew,  §§85, 
86,  89;  Kimball,  §§127,  141;  Reed  &  Guthe,  §§54,  55;  Spinney,  §51;  Watson, 
§§112-114;  Watson  (Pr.),  §§46-49. — More  Advanced:  McLean's  Practical 
Physics,  §88;  Poynting  &  Thomson,  (Properties  of  Matter)  Chap.  II. 

The  acceleration  of  gravity,  g,  is  most  readily  obtained  from  the 
length  and  time  of  vibration  of  a  pendulum.  The  time  of  vibra- 
tion of  an  ideal  simple  pendulum,  i.  e.,  a  heavy  particle  vibrating 
at  the  end  of  a  massless  cord  would  be 


I  being  the  length  of  the  pendulum.  If  the  bob  is  a  ball  so  large 
that  the  mass  of  the  suspending  wire  is  negligible,  the  above 
formula  will  apply  provided  the  radius  of  the  ball  is  negligible 
compared  with  the  length  of  the  pendulum.  If  these  assumptions 
may  not  be  made,  the  pendulum  must  be  regarded  as  a  physical 
pendulum  and  its  moment  of  inertia  about  the  suspension  con- 
sidered. Under  these  circumstances  the  formula 


Mgh 

must  be  used,  where  /  is  the  moment  of  inertia  of  the  entire 
pendulum  about  the  knife-edge,  M  is  the  total  mass  and  h  is 
the  distance  from  the  knife-edge  to  the  center  of  gravity  of  the 
3 


34  MECHANICS. 

whole.  If  the  mass  of  the  suspension  is  negligible  it  is  only 
necessary  to  consider  the  moment  of  inertia  of  the  ball  about  the 
knife-edge.  It  is  easily  shown  that  the  latter  formula  then  re- 
duces to  the  formula  for  the  simple  pendulum,  provided  the 
length  of  the  pendulum  is  taken  as  the  distance  from  the  knife- 
edge  to  the  center  of  the  ball  plus  2r2/$l  where  r  is  the  radius  of 
the  ball.  Hence  to  find  g  there  are  three  quantities,  t,  /,  and  r, 
to  be  measured. 

A  convenient  form  of  pendulum  consists  of  a  spherical  bob 
into  which  screws  a  nipple  through  which  a  fine  wire  is  passed 
and  secured.  To  the  upper  end  of  the  wire  is  soldered  a  stirrup 
of  brass  which  rests  on  a  knife-edge  of  steel.  A  short  platinum 
wire  should  be  soldered  to  the  lower  side  of  the  bob. 

For  accurately  measuring  the  length  of  the  pendulum  a  catheto- 
meter  (see  p.  17),  which  should  be  carefully  adjusted,  may  be 
used.  (If  necessary,  the  measurement  of  length  may  be  post- 
poned until  the  time  has  been  observed).  The  horizontal  cross- 
hair of  the  cathetometer  is  first  focused  on  the  knife-edge,  the 
fine  screw  being  used  for  the  final  adjustment  of  the  telescope, 
and  the  scale  and  vernier  are  then  read.  The  telescope  is  then 
lowered  and  set  on  either  the  top  or  bottom  of  the  bob,  which- 
ever is  the  more  definite.  These  readings  should  be  repeated 
several  times,  beginning  each  time  with  the  knife-edge.  If  the 
adjustments  are  imperfect,  the  telescope  should  at  least  be  made 
exactly  level  before  each  reading.  The  diameter  of  the  bob  may  be 
measured  by  means  of  a  micrometer  or  a  vernier  caliper  (see  p.  13). 

For  fixing  the  vertical  position  of  the  pendulum,  two  vertical 
pointers  may  be  so  placed  that,  when  the  pendulum  is  at  rest, 
the  pendulum  suspension  and  two  pointers  are  in  one  plane.  The 
eye  of  the  observer  should  always  be  kept  in  this  plane  in  using 
the  first  two  methods.  The  pendulum  is  set  vibrating  in  an  arc 
of  3  or  4  cms.  Several  attempts  may  be  necessary  to  get  the 
pendulum  vibrating  exactly  perpendicular  to  the  knife-edge 
with  the  bob  free  from  rotation. 

The  time  of  vibration  is  most  readily  obtained  with  precision 
when  the  pendulum  is  very  nearly  a  second's  pendulum,  i.  e., 
when  the  period  of  a  complete  vibration  is  very  nearly  two 
seconds.  For  the  determination  of  the  period  several  methods 


ACCELERATION   OF   GRAVITY   BY   PENDULUM.  35 

are  available.     The  first  and  roughest  method  given  below  will 
serve  for  adjusting  the  pendulum  to  the  required  length. 

(A)  In  the  first  method  for  determining  the  period,  time  is 
found  by  the  relay  (p.  23)  and  the  number  of  vibrations  in  three 
minutes  is  counted,  fractions  of  a  vibration  being  roughly  esti- 
mated.    This  is  repeated  several  times.     Or  a  stop-watch  or  stop- 
clock  may  be  used,  but  it  should  be  rated  by  comparison  with  a 
chronometer  or  standard  clock.     The  stop-watch  is  started  as 
the  pendulum  crosses  the  plane  of  observation  and   "one"  is 
counted  the  next  time  the  pendulum  crosses  the  plane  in  the  same 
direction.     The  watch  is  stopped  on  the  5Oth  vibration,  and  the 
whole  repeated  five  times.     The  mean  time  divided  by  50  will 
give  a  fair  value  for  the  period. 

(B)  A  second  and  much  more  accurate  method  of  obtaining 
the  time  of  vibration  is  the  method  of  coincidences.     This  consists 
in  finding  the  rate  at  which  the  pendulum  gains  or  loses  as  com- 
pared with  a  standard  clock  or  chronometer.     It  is  applicable 
only  when  the  periods  of  pendulum  and  clock  or  chronometer 
are  nearly  the  same  or  when  one  is  nearly  an  exact  multiple  of 
the  other.     The  method  receives  its  name  from  the  fact  that 
what  is  observed  is  the  "coincidence  interval"  or  the  interval 
between  the  moment  when  a  passage  of  the  pendulum  through  the 
vertical  coincides  with  some  signal  from  the  clock  to  the  next 
time  when  such  a  coincidence  occurs. 

In  a  coincidence  interval,  the  pendulum  must  gain  or  lose  one 
vibration  as  compared  with  the  chronometer  or  other  time 
standard.  If  n  such  coincidence  intervals  occur  in  T  sec.,  the 
number  of  vibrations  of  the  pendulum  during  this  time  is  (T  =«=  n). 
Hence  if  t  is  the  time  of  one  vibration, 

T 
*- 


and  the  period  of  a  complete  vibration  is 

t- 

V  - 


A  convenient  form  of  signal  is  given  by  the  chronometer  and 
relay  described  on  page  23.     It  is  advisable  to  have  the  coinci- 


36  MECHANICS. 

dence  interval  something  between  30  seconds  and  3  minutes, 
and,  if  necessary,  the  length  of  the  pendulum  should  be  changed 
for  the  purpose. 

After  the  coincidence  interval  has  been  roughly  determined 
by  a  few.  observations,  the  following  modification  of  the  method 
will  give  it  much  more  accurately.  Calling  the  time  of  the  first 
coincidence  zero  seconds,  observe  the  second  on  which  the  next 
coincidence  occurs  and  then  the  next,  until  four  have  been  ob- 
served. Then,  after  allowing  a  considerable  number  of  coinci- 
dences to  pass  unnoted,  but  keeping  note  of  the  time,  observe  the 
number  of  the  seconds,  counted  from  the  original  coincidence, 
upon  which  four  more  successive  coincidences  occur. 

From  the  first  set  of  coincidences,  three  estimates  of  the  coinci- 
dence interval  will  be  obtained  and  three  others  from  the  second 
set,  the  mean  of  all  giving  an  approximate  estimate.  Then  let 
the  time  of  the  first  coincidence  of  the  first  set  be  subtracted  from 
the  time  of  the  first  of  the  second  set,  also  the  time  of  the  second 
coincidence  of  the  first  set  from  that  of  the  second  of  the  second 
set,  etc.  These  differences  give  four  estimates  of  the  time,  T, 
of  some  unknown  integral  number,  n,  of  coincidence  intervals. 
If  the  mean  of  these  four  estimates  be  divided  by  the  mean  time 
of  a  single  coincidence  interval  as  already  found,  the  quotient 
will  be  n  plus  or  minus  a  small  fraction.  This  fraction  is  due  to 
inaccuracy  in  the  estimates  of  the  coincidence  intervals  and 
should  be  dropped.  The  period  /  of  the  pendulum  may  now  be 
calculated.  The  plus  sign  in  the  denominator  is  used  if  the 
pendulum  is  the  faster. 

The  following  aid  to  the  observation  of  coincidences  is  sug- 
gested. Keeping  the  eye  constantly  in  the  proper  plane  for  ob- 
servation, make  a  dot  on  a  piece  of  paper  at  each  click  of  the 
relay.  When  there  appears  to  be  coincidence,  prolong  the  dot 
into  a  stroke.  To  avoid  recording  every  click,  a  cross  may  be 
used  instead  of  a  dot  for  marking  a  minute,  and  the  clicks  may  be 
passed  unrecorded  until  the  next  minute,  or  coincidence.  There 
may  be  several  successive  clicks  during  which  there  appear  to  be 
coincidences,  in  which  case  several  successive  strokes  should  be 
made  and  the  mean  taken.  From  these  dots,  strokes,  and  crosses, 
the  times  of  coincidence  may  be  deduced.  Or,  a  dial  indicating 


COEFFICIENT   OF   FRICTION.  37 

seconds  may  be  employed,  the  second  when  there  first  appears  to 
be  a  coincidence  being  observed  and  the  second  when  there  first 
appears  to  be  no  coincidence.  Since  the  clock  cannot  be  ob- 
served immediately,  the  ticks  are  counted  until  the  clock  is  ob- 
served and  then  subtracted;  minutes  must  be  noted  and  re- 
corded if  they  are  not  recorded  on  the  clock. 

(C)  A  third  method  consists  in  modifying  the  second  method 
so  that  coincidences  of  two  sounds  are  observed.  The  pendulum 
is  made  to  actuate  a  sounder  or  telephone  each  time  it  passes 
through  the  vertical  and  a  coincidence  is  observed  when  the 
sounder  and  relay  strike  together.  A  block  of  wood  with  a  nar- 
row trough  filled  full  of  mercury  is  placed  in  a  mercury  tray  and 
is  adjusted  beneath  the  pendulum  so  that  the  platinum  wire  on 
the  under  side  of  the  bob  just  touches  the  mercury  when  the 
pendulum  is  at  rest,  and  crosses  the  narrow  trough  at  right  angles 
when  the  pendulum  is  in  motion.  A  wire  soldered  to  the  knife- 
edge  is  connected  in  series  with  several  batteries,  a  sounder  or 
telephone,  and  the  mercury  trough.  The  final  adjustment  of 
the  mercury  trough  is  made  with  the  leveling  screws  of  the  mer- 
cury tray.  Care  should  be  taken  not  to  spill  the  mercury. 

From  the  possible  errors  in  nie  measurements  of  /  and  /  deduce 
the  possible  errors  in  the  value  found  for  g  (see  pages  4-7). 

Questions. 

1.  Does  the  friction  of  the  knife-edges  and  of  the  air  increase  or  decrease 
the  value  of  g? 

2.  Why  should  coincidence  be  observed  exactly  for  the  plane  containing  the 
position  of  rest? 

3.  What  would  be  the  result  of  increasing  the  arc  of  vibration  to  10  cm.? 
(Table  III.) 

4.  Why  should  the  top  reading  of  the  cathetometer  always  precede  the 
bottom  reading? 

5.  Design,  if  possible,  a  scheme  of  electrical  connections  such  that'the  sounder 
will  only  operate  when  there  is  a  coincidence. 

VIII.    COEFFICIENT  OF  FRICTION. 

References — Elementary:  Duff,  §§126-130;  Ames,  p.  118;  Crew,  §117;  Daniell's 
Textbook  of  the  Principles  of  Physics,  pp.  176-184;  Kimball,  §§81-82;  Reed 
&  Guthe,  §48;  Spinney,  §§73-75;  Watson,  §§96-100. — More  Advanced: 
McLean's  Practical  Physics,  §§93-95. 

The  coefficient  of  friction  of  two  surfaces  is  the  ratio  of  the  force 
of  friction  opposing  the  incipient  or  actual  -relative  motion  to 


38  MECHANICS. 

the  force  pressing  the  two  surfaces  together.  The  force  requisite 
to  start  the  motion  is  greater  than  that  required  to  sustain  the 
motion,  i.  e.,  the  "coefficient  of  static  friction"  is  greater  than 
that  of  "  kinetic  friction."  Moreover,  the  coefficient  of  kinetic 
friction  is  not  quite  constant,  but  varies  somewhat  with  the  speed. 

(A)  The  coefficient  of  static  friction  of  one  surface  on  another 
may  be  found  by  means  of  a  block  of  the  former  resting  on  a  slide 
of  the  latter.     One  end  of  the  slide  is  gently  elevated  by  a  screw 
until  the  block  just  fails  to  stand  stationary  on  the  slide.     The 
tangent  of  the  angle  which  -the  slide  then  makes  with  the  hori- 
zontal  equals   the   coefficient  of  static   friction    (see   references). 
The  tangent  may  be  measured  by  some  simple  method,  using 
meter-stick,  plumb-line   and   level   or  square.     Several  entirely 
independent  adjustments  for  this  angle  and  measurements  of  the 
tangent  should  be  made,  the  adjusting  screw  being  each  time 
turned  some  distance  down  so  that  the  influence  of  the  previous 
setting  may  be  avoided.     The  friction  may  vary  somewhat  from 
point  to  point,  and  if  so,  different  points  should  be  chosen  for  the 
separate  trials. 

The  accuracy  of  the  determination  of  the  tangent  should  be 
calculated  to  see  whether  the  possible  errors  will  account  for  the 
variations  of  the  coefficient.  Such,  however,  will  probably  not 
be  found  to  be  the  case. 

(B)  The  coefficient  of  kinetic  friction  may  be  determined  by  the 
same  apparatus  if  we  can  find  the  acceleration  with  which  the 
block  moves  down  the  slide  when  the  latter  is  tilted  beyond  the 
angle  of  repose.     For,  if  the  acceleration  of  the  block  is  a  and 
its  mass  m  and  the  angle  of  inclination  of  the  slide  i,  then  the 
component  of  gravity  down  the  slide  is  mg  sin  i  and  the  pressure 
on  the  slide  is  mg  cos  i.     Hence,  if  ^  is  the  coefficient  of  friction, 
by  Newton's  second  law, 


and, 


m  a  —  m  g  sn    —  ^mg  cos 
a 


g  COS 


This  process  will  give  the  mean  coefficient  of  friction  for  the 
range  of  speeds  through  which  the  block  passes,  but  for  the  low 
speeds  in  question  the  coefficient  does  not  vary  much. 


COEFFICIENT   OF   FRICTION.  39 

The  acceleration,  a,  is  found  by  a  method  frequently  employed 
in  physical  measurements.  A  tuning-fork  (frequency  of  50  or 
less)  is  fastened  in  a  clamp  attached  to  a  support  above  the  slide. 
A  stylus  of  spring  brass  with  a  steel  needle  point  is  attached  to 
one  prong  and  just  behind  this  stylus  is  a  second  stationary  stylus 
which  is  attached  to  the  support.  A  long  and  narrow  glass  plate 
is  covered  with  the  washing  compound  called  "Bon  Ami"  by 
transferring,  with  a  wet  cloth,  a  little  of  the  paste  from  the  cake 
to  the  glass,  and  then  spreading  it  out  in  a  thin  layer.  The 
block  is  then  raised  to  the  top  of  the  slide  and  secured  by  a  trigger. 
The  support  that  carries  the  fork  is  raised  and  lowered  and  the 
fork  is  adjusted  in  the  clamp  until  each  stylus  touches  the  coated 
glass,  making  with  it  an  angle  of  about  45°,  the  stylus  on  the  fork 
being  exactly  in  front  of  the  other  stylus. 

The  frame-work  is  then  lifted  until  neither  stylus  touches  the 
glass.  The  fork  is  set  in  vibration  by  drawing  the  prongs  to- 
gether with  the  fingers  and  releasing  them,  or  by  withdrawing  a 
wooden  wedge,  and  is  then  adjusted  until  its  stylus  vibrates  an 
equal  amount  on  each  side  of  the  other  (stationary)  stylus. 
The  frame-work  is  then  lowered  until  the  styli  touch  the  glass 
and  the  block  is  immediately  released  by  the  trigger. 

A  wave  line  should  be  obtained  with  a  straight  line  exactly  in 
the  center,  the  amplitude  of  the  wave  line  on  each  side  of  the 
straight  line  being  several  millimeters.  Since  in  any  measure- 
ment the  effect  of  inaccuracies  at  the  ends  is  less  important  the 
greater  the  quantity  measured,  we  measure  the  distance  passed 
over  during  several  vibrations  of  the  fork.  This  distance  divided 
by  the  time  in  which  it  was  traversed,  i.  e.,  by  the  period  of  the 
fork  multiplied  by  the  number  of  vibrations,  gives  the  average 
velocity  of  the  block  during  this  time.  If  T  be  the  period  of  the 
fork  and  x  the  distance  passed  over  in  n  complete  vibrations  of  thQ 
fork,  the  average  velocity  is  x/nT.  Similarly  we  find  the  average 
velocity  for  the  next  n  complete  vibrations.  The  average 
acceleration  will  be  the  difference  between  these  average  velocities 
divided  by  their  separation  in  time  or  n  T;  for  since  each  velocity 
is  the  average  we  may  consider  it  as  belonging  to  the  middle  of 
the  time  for  which  it  is  the  average.  From  several  successive 


40  MECHANICS. 

groups  of  n  vibrations  several  values  of  the  acceleration  are  ob- 
tained and  the  mean  taken. 

It  remains  to  determine  the  period  of  the  fork.  Two  methods 
will  be  described,  (a)  The  fork  is  clamped  beside  a  small  electro- 
magnet connected  through  a  battery  with  a  pendulum  which 
closes  the  circuit  every  second  (see  p.  23).  To  the  armature  of 
the  electro-magnet  a  stylus  is  also  attached.  A  plate  of  glass 
covered  with  "Bon  Ami"  is  clamped  on  a  movable  block  so  that 
each  stylus  rests  upon  it.  The  electro-magnet  and  fork  may  have 
any  relative  position  which  may  be  convenient,  but  the  styli 
should  not  be  far  apart.  The  fork  is  set  vibrating  and  the  block 
with  the  glass  is  drawn  along,  the  fork  making  a  wave  line  and 
the  other  stylus  a  straight  line  broken  (or  notched)  every  second. 
With  a  square,  lines  are  drawn  at  right  angles  to  the  glass  through 
the  beginnings  of  alternate  second  signals  and  the  number  of 
complete  vibrations,  estimated  to  tenths,  is  counted  between 
the  lines. 

(b)  This  is  known  as  a  stroboscopic  method  and  depends  upon 
the  persistence  of  vision.  The  fork  is  watched  through  holes  in  a 

disk  revolving  at  a  constant  speed. 
The  holes  are  equally  spaced  in  con- 
centric circles,  the  number  per  circle 
increasing  with  the  radius.  The  speed 
of  the  disk  is  varied  until  the  fork 
appears  stationary  when  viewed 
through  the  holes  of  a  particular 
FIG.  9.  circle.  If  there  are  m  holes  in  the 

circle,    and    if    the    disk    revolves    n 

times  per  second,  the  frequency  of  the  fork  is  obviously  mn.  By 
varying  the  speed  and  using  other  holes,  additional  determina- 
tions may  be  made.  The  speed  of  the  disk  is  obtained  by  de- 
termining, with  a  counter,  the  number  of  revolutions  in  a  given 
time. 

(C)  Another  method  of  finding  the  coefficient  of  kinetic  friction 
is  to  make  the  slide  horizontal  and  find  the  force  required  to  keep 
the  block  in  uniform  motion  after  it  has  been  started.  For  this 
purpose  a  braided  cord  is  attached  to  one  end  of  the  block, 
passed  over  a  pulley  at  the  end  of  the  slide,  and  attached  to  a  scale 


COEFFICIENT   OF   FRICTION.  4! 

pan,  to  which  weights  are  added.  In  this  case  the  weight  of  the 
pan  and  weights  must  not  be  taken  as  the  force  acting  on  the 
block,  for  some  force  is  required  to  overcome  the  friction  of  the 
pulley.  The  amount  required  must  be  found  by  a  separate  ex- 
periment. Two  pans  are  attached  to  the  ends  of  the  cord  hanging 
over  the  pulley  and  sufficient  equal  weights  are  placed  on  the 
pans  to  make  the  pressure  on  the  pulley  the  same  as  in  the  main 
experiment  where  the  parts  of  the  cord  were  at  right  angles. 
The  additional  weight  on  one  scale  pan  requisite  to  keep  the  whole 
in  constant  motion  when  started  is  the  force  needed  to  overcome 
the  friction  of  the  pulley,  and  is,  therefore,  the  correction  required. 

With  this  apparatus  we  may  also  test  whether  the  coefficient 
of  friction  varies  when  weights  are  added  to  the  block.  The 
correction  for  friction  of  the  pulley  does  not  need  to  be  re-de- 
termined experimentally,  but  may  be  calculated  from  the  former 
determination,  on  the  assumption  that  the  friction  of  the  pulley 
is  proportional  to  the  pressure  on  it. 

The  possible  error  in  the  results  of  the  first  and  last  methods  is 
easily  determined.  The  most  accurate  way  of  finding  the  possi- 
ble error  in  method  (B)  is  by  means  of  formulae  deduced  by  the 
differential  calculus  (see  p.  6),  but  a  much  simpler  and  a  suffi- 
ciently accurate  method  is  the  following:  Note  that  an  over- 
estimate of  i  will  increase  the  value  of  /z  and  the  same  will  be  the 
effect  of  an  underestimate  of  a.  Hence  the  coefficient  should  be 
recalculated  with  tan  i  and  cos  i  increased  and  a  decreased  by 
their  possible  errors  and  the  change  found  in  the  coefficient  may 
be  taken  as  the  final  possible  error.  The  possible  error  of  a 
may  be  taken  as  its  mean  deviation  and  the  possible  errors  of 
tan  i  and  cos  i  may  be  deduced  from  the  measurements  from 
which  they  were  obtained. 

Questions. 

1.  In  the  second  method,  why  is  it  desirable  that  the  straight  line  be  exactly 
in  the  middle  of  the  wave  line? 

2.  In  the  third  method,  what  error  would  be  introduced  if  the  cord  from  the 
block  was  not  exactly  horizontal? 


42  MECHANICS. 


IX.  HOOKE'S  LAW  AND  YOUNG'S  MODULUS. 

References—  Elementary:  Duff,  §§168,  171,  173;  Ames,  pp.  144,  145,  153,  154; 
Crew,  §§126-129;  Kimball,  §§238-244;  Reed  &  Guthe,  §§57-59;  Spinney, 
§§92-96;  Watson,  §§172-3.  —  More  Advanced:  McLean's  Practical  Physics, 
§§112-115;  Poynting  &  Thomson  (Prop,  of  Matter),  §§73-78,  85-102; 
Tail's  Properties  of  Matter,  Chap.  VIII. 

Hookes  Law  states  that,  for  small  strains,  stress  and  strain 
are  proportional.  Young's  Modulus,  E,  is  the  constant  ratio 
of  stress  to  strain  for  a  stretching  strain,  the  stress  being  taken 
as  the  force  per  unit  cross  section  and  the  strain  as  the  stretch 
per  unit  of  length,  or,  if  F  is  the  whole  force,  A  the  area  of  cross 
section,  L  the  whole  length,  and  /  the  increment  of  length, 


=-= 

A  '  L     Al' 

(A)    The  quantity  most  difficult  to  measure  is  /,  the  small 
increase  of  length.     If  a  wire  be  supported  at  one  end  and  force 
applied  to  the  other  end,  there  is  danger  that  the  support  may 
yield  slightly,  and  a  slight  amount  of  yielding  will 
cause  a  proportionally  large  error  in  the  estimate  of  the 
small  increase  in  length.     The  peculiarity  of  the  first 
method   described   below   is   the   means   adopted   to 

J  eliminate  the  yield  of  the  support.  The  increase  of  the 
length  of  the  wire  under  experiment  is  found  by  com- 
parison with  another  wire  under  constant  stretch 
attached  to  the  same  support  as  the  former  wire. 
One  wire  carries  a  scale  and  the  other  a  vernier  oppo- 
site the  scale.  If  there  be  any  doubt  which  is  vernier 
(see  p.  12)  and  which  is  scale,  comparison  should  be 
made  with  an  ordinary  steel  scale.  The  screws  by 
means  of  which  the  wires  are  clamped  to  scale  and 
F  vernier  should  be  adjusted  until  scale  and  vernier 

tend  to  lie  in  one  plane.     A  light  rubber  band  may 

then  be  slipped  over  scale  and  vernier  to  keep  them  together. 
The  stretch  may  be  produced  by  means  of  lead  weights.     The 

value  of  these  weights  should  be  determined  by  a  platform  balance. 

To  produce  a  suitable  stretch  it  may  be  advisable  to  add  two  or 


HOOKE  S  LAW  AND  YOUNG  S  MODULUS.          43 

more  weights  at  a  time.  We  shall  suppose  that  two  are  added, 
but  the  description  can  readily  be  modified  to  suit  any  number. 
The  greatest  weight  should  not  be  more  than  half  that  required 
to  break  the  wire.  (A  copper  wire  o.oi  sq.  cm.  section  will 
break  at  40  kgs.;  brass,  60  kgs.;  iron,  60  kgs.)  Suppose,  then, 
two  weights  are  added  at  a  time  and  each  stretch  observed. 
When  the  maximum  number  has  been  added  the  weights  should 
be  removed  in  the  same  order,  readings  being  again  taken  as  they 
are  removed.  The  whole  series  of  observations  should  be  re- 
peated at  least  three  times.  Such  readings  should  always  be 
arranged  in  tables  having  in  a  line  or  column  all  the  readings 
for  a  particular  pair  of  weights.  The  length  of  the  wire  may  be 
measured  by  means  of  a  long  beam  compass  and  the  diameter 
should  be  measured  at  least  a  dozen  times  at  different  places  and 
in  different  directions*  by  means  of  a  micrometer  caliper  (see 

P-  13). 

Before  calculating,  the  dimensions  should  be  expressed  in 
centimeters  and  the  weights  in  dynes.  First  find  the  mean  value 
of  /  for  each  pair  of  weights  when  added  and  when  removed  and 
then  the  value  of  F  -f-  /  for  each  of  these  values  of  /  and  the  re- 
spective F's.  Find  the  mean  value  of  F  -r-  /  and  the  greatest 
percentage  deviation  from  the  mean.  This  will  give  the  per- 
centage deviation  from  Hooke's  Law  since  F  -f-  /  should  be  a 
constant,  A  and  L  being  practically  constant.  The  final  value 
of  Young's  Modulus  should  be  stated  in  the  notation  explained 
on  page  10. 

The  possible  errors  of  the  different  quantities  measured  may 
be  taken  as  the  mean  deviation  in  each  case.  The  percentage 
error  of  the  final  value  of  R  will  be,  as  is  readily  seen  from  the 
formula,  the  sum  of  the  percentage  errors  of  F,  L,  /,  and  twice  the 
percentage  error  of  the  radius  (see  p.  5). 

(B)  Young's  Modulus  may  also  be  found  by  means  of  the 
flexure  of  a  bar.  For,  in  bending  (within  limits)  one  side  of  a  bar 
is  stretched  and  the  other  compressed  (negatively  stretched), 
and  so  Young's  Modulus  is  the  only  constant  that  need  be  con- 
sidered. The  amount  of  bending  might  be  deduced  from  the 
sag  of  the  center  or  end  of  the  bar,  but  a  much  more  delicate 
method  is  the  following  optical  one: 


44 


MECHANICS. 


A  bar  of  rectangular  cross  section  is  laid  on  two  knife-edges 
and  at  each  end  is  attached  an  approximately  vertical  mirror  in 
mountings  that  admit  of  considerable  adjustment.  A  vertical 
scale,  nearly  in  line  with  the  bar,  is  reflected  from  the  farther 
mirror  into  the  nearer  and  thence  into  a  telescope  also  nearly 
in  line  with  the  bar.  When  a  weight  is  attached  to  the  center  of 
the  bar,  the  bar  is  bent  and  another  part  of  the  scale  is  reflected 
into  the  telescope.  This  arrangement  serves  to  determine  the 
angle  of  bending.  For  suppose  the  difference  of  the  scale- 
readings  on  the  horizontal  cross-hairs  of  the  telescope  be  D  cms. 
(Fig.  u)  and  let  the  distance  between  the  two  mirrors  be  p  and 
the  distance  of  the  scale  from  the  farther  mirror  q,  then,  if  the 
change  of  inclinat'on  of  each  mirror  be  i, 


D 


tan  i  = 


For  a  consideration  of  figure  n  will  show  that  di=p  tan  2i;  d2  = 
But  since  i  is  a  small  angle  tan  2i=2  tan  i  and  tan  42  =  4  tan  i. 


tan 


tan 


FIG.  ii. 

From  tan  i,  the  weight  R  in  dynes  applied  at  the  center,  the 
length  of  the  bar  between  the  knife-edges,  /,  the  breadth,  b, 
and  the  thickness,  a,  Young's  Modulus,  E,  is  obtained,  by  the 
equation 

RP 


PROOF. 

Let  the  y  axis  coincide  with  the  radius  from  the  center  of  curvature  of  the 
bar  to  the  center  of  the  bar,  and  let  the  x  axis  be  the  tangent  to  the  central  axis 
at  this  point.  Designate  distances  from  the  elastic  central  axis,  LOM  (Fig.  12) 
along  other  radii  by  z.  The  elastic  central  axis  remains  unchanged  in  length. 
The  curvature  at  any  point  P  on  LOM  is  the  rate  of  change  of  the  directions 


HOOKE'S  LAW  AND  YOUNG'S  MODULUS. 


45 


of  the  tangent.  The  angle  the  tangent  line  at  P  makes  with  the  x-axis  is  a 
small  one  and  may  be  taken  as  dy/dx  (which  is  really  the  tangent  of  that  angle). 
The  rate  of  change  of  the  direction  of  the  curve  at  the  point  x,  y,  is  therefore 
d2y/dx2,  which  therefore  equals  the  curvature.  But  the  curvature  also  equals 
i/r,  r  being  the  radius  of  curvature.  Hence 

i  _dzy 
~r~dx?' 

Now  consider  two  strips  of  the  beam  distant  ±2  from  LOM.     By  the  bending 

d2y 
these  strips  arc   changed   in   length   in   the   proportion  z/r  or  z  -j^.     (For, 

consider  figure  13;  the  proportional  change  of  length  is 
G'H'-GH    G'C-GC 


GH 


GC 


GG'=A 
GC      r) 


FIG.  12. 


FIG.  13. 


By  definition  of  Young's  Modulus,  if  a  force  F  applied  to  a  rod  of  cross  section 
A  and  length  L  produce  an  extension  /, 

EAl 


F= 


L  ' 


where  E  is  Young's  Modulus.     The  stress  in  a  strip  of  width  b  and  thickness 
dz  is  obtained  by  putting  bdz  for  A  and  z  d*y/dx2  for  l^-L,  which  gives 

„,  ,     d2? 
£002.23-;-. 
d*2 

Hence  the  moment  about  P  of  the  restoring  force  in  the  strips  =*=  2  is 


.. 
dx2 

The  moment  about  P  of  the  stress  in  the  whole  cross  section  is  the  integral  with 
reference  to  z  of  the  above  expression  for  values  of  z  from  o  to  ^a  or 


12 


For  equilibrium  this  must  equal  the  moment  of  }4R  about  P  or  j^R  (}4l  —  x) 


_ 

"  dx2~ Easb  \  2 
.  dy=6R    I  lx_x2 
'''dx~Ea?b  1    2      2 


46  MECHANICS. 

At  a  point  of  support 

dy 
,    x  =  J^l;  dx  =  tan  i. 

Hence  by  substitution 

F     V      ™ 
E-A' 


(By  integrating  again,  the  value  of  y  at  a  point  of  support  or  the  deflection  of 
the  beam  is  obtained.     This  is  left  as  an  exercise  for  the  student.) 

The  adjustment  of  the  apparatus  is  most  readily  made  as  fol- 
lows. Place  the  telescope  and  scale  nearly  in  the  line  of  the  mir- 
rors and,  glancing  above  the  telescope,  set  the  farther  mirror  so 
that  the  nearer  mirror  is  seen  by  reflection  and  then  the  latter  so 
that  the  scale  is  seen.  Then  adjust  the  eye-piece  of  the  telescope 
so  that  the  cross-hairs  are  as  distinct  as  possible  and  finally  focus 
the  telescope  until  the  scale  is  seen.  The  bar  must  not  be  strained 
beyond  the  limits  of  elasticity.  For  further  details  of  these 
adjustments,  see  p.  23.  Equal  weights,  perhaps  100  grams  at  a 
time,  should  be  added,  but  this  process  should  be  stopped  when 
it  is  found  that  the  scale-reading  no  longer  changes  in  the  same 
proportion  as  the  weights.  Determine  carefully  by  several 
readings,  with  and  without  this  maximum  weight  attached,  the 
change  of  scale-reading.  The  width  and  thickness  of  the  bar 
may  be  measured  by  a  micrometer  caliper  (see  p.  13),  a  number 
of  readings  of  each  at  different  points  being  made. 

In  calculating,  use  this  weight  for  R  and  the  average  of  the 
changes  of  deflection  for  D  and  take  the  mean  deviation  as  the 
measure  of  the  possible  error  of  D,  a,  and  b.  The  percentage 
possible  error  of  tan  i  is  deduced  from  the  possible  errors  of  D 
and  2p  +  4<Z-  The  possible  error  in  the  latter  term  is  twice  the 
possible  error  in  p  plus  four  times  that  in  q. 

(C)  A  simple  optical  method  may  also  be  employed  for  finding 
the  extension  of  a  wire.     In  this  method,  one  side  of  a  small 
bench  carrying  a  vertical  mirror  is  supported  by  the  end  of  the 
wire  and  the  other  by  a  fixed  bracket.     The  deflection  of  the 
mirror  when  weights  are  added  to  the  wire  is  read  by  a  scale  and 
telescope.     The  details  of  the  method  may  readily  be  worked  out 
by  anyone  who  has  followed  the  preceding  methods. 

(D)  (Searle's   Method.)     The  extension  may  also  be  deter- 
mined from  the  change  of  position  of  a  level  supported  by  the  two 


RIGIDITY.  47 

wires.  The  lowering  of  the  stretched  wire  is  compensated  by  a 
micrometer  screw  which  therefore  reads  the  extension.  For  de- 
tails, see  Searles  Experimental  Elasticity,  §48,  or  Watson's 
Practical  Physics,  §45. 

Questions. 

1.  How  closely  is  it  worth  while  to  measure  the  length  of  the  wire  in  the 
first  method? 

2.  Which  of  the  first  two  methods  is  the  more  accurate  and  what  is  the  chief 
weakness  of  the  other? 

3.  In  the  second  method  why  is  nothing  said  as  to  the  distance  of  the 
mirrors  beyond  the  knife-edges?     Might  they  be  placed  inside? 

4.  Reduce  your  results  to  tons  and  inches. 

X.    THE  RIGIDITY  (OR  SHEAR-MODULUS). 

References — Elementary:  Duff,  §§119,  169-170;  Duff's  Mechanics,  §§117, 
130-131;  Ames,  pp.  151-153;  Crew,  §§131,  132;  Kimball,  §243;  Watson, 
§§171,  174-175. — More  Advanced:  McLean's  Practical  Physics,  §§116-121; 
Poynting  &  Thomson  (Prop,  of  Matter},  §§78-84;  Tail's  Properties  of 
Matter,  Chap.  XL 

The  rigidity  of  any  material  is  the  resistance  it  offers  to  change 
of  shape  without  change  of  volume.  It  is  measured  by  the  ratio 
of  the  shearing  stress  to  the  shear  produced.  In  the  twisting  of 
a  wire  or  rod,  within  moderate  limits,  there  is  no  change  of  vol- 
ume. Hence  this  affords  a  means  of  finding  the  rigidity  of  the 
material.  The  constant  or  modulus  of  torsion  of  a  particular 
wire  is  the  couple  required  to  twist  one  end  of  unit  length  of  the 
wire  through  unit  angle,  the  other  end  being  kept  fixed.  If  it 
be  denoted  by  r  and  if  the  length  of  the  wire  be  L  the  couple 
required  to  twist  the  wire  through  unit  angle  is  r/L.  If  now  to 
the  wire  be  attached  a  mass  of  moment  of  inertia,  /,  and  the  wire 
and  the  mass  be  set  into  torsional  vibrations,  the  time  of  a  semi- 
vibration  is,  by  the  principles  of  Simple  Harmonic  Motion  (see 
references) , 


If  /,  L  and  /  be  found  r  can  be  deduced.  From  the  modulus 
of  torsion  of  the  particular  wire  the  rigidity  n  of  the  material  of 
which  the  wire  consists  can  be  deduced ;  for 

2r 


48  MECHANICS. 

Proof. 

Suppose  unit  length  of  the  wire  to  be  twisted  through  unit  angle.  The 
vibrations  are  due  to  the  restoring  couple  at  the  lower  end  produced  by  the 
twist.  Let  the  cross  section  of  the  end  be  divided  into  concentric  rings  and 
let  the  radius  of  one  ring  be  x  and  its  width  dx;  its  area  is  2irxdx.  Relatively 
to  the  fixed  end  it  is  displaced  through  unit  angle.  Hence  the  linear  displace- 
ment (supposed  small)  of  the  ring  whose  radius  is  x  is  x  times  unit  angle  or 
simply  x.  This  is  by  definition  the  shear  and  hence  the  shearing  stress  is  nx. 
This  is  the  restoring  force  per  unit  area  of  the  cross  section.  Hence  the  re- 
storing force  of  the  ring  whose  radius  is  x  is  2wnx^dx.  The  effect  of  this  force 
in  producing  rotation  depends  on  its  moment  about  the  axis  or  2irnx3dx.  The 
moment  of  the  restoring  force  of  the  whole  end  section  is  the  sum  of  expressions 
like  2inixzdx  for  value  of  x  between  o  and  r  or  y^irnr*.  This  is  by  definition  the 
modulus  of  torsion,  r  and  gives  us  the  above  equation.  It  should  be  noticed 
that  it  is  not  a  constant  for  the  material  of  the  wire,  but  depends  on  the  dimen- 
sions of  the  particular  wire. 

The  length,  L,  may  be  measured  by  means  of  a  long 
compass  which  is  afterward  compared  with  a  fixed  brass  scale. 
The  radius,  R,  may  be  measured  by  a  micrometer  caliper  (see 
p.  13),  measurements  being  made  at  a  great  many  different 
places  and  the  mean  taken. 

The  moment  of  inertia,  /,  of  the  wire  and  attached  mass  might 
be  roughly  obtained  by  calculation,  but  it  is  better  to  apply  an 
experimental  method  that  is  used  in  other  cases.  This  consists 
in  adding  to  the  vibrating  mass,  of  unknown  moment  of  inertia, 
another  mass  of  such  regular  form  that  its  moment  of  inertia 
can  be  accurately  calculated,  and  finding  tjie  times  of  vibration 
before  (/)  and  after  (T)  adding  this  mass.  If  the  original  moment 
of  inertia  be  /  and  the  added  moment  of  inertia  i: 


t  •    Vi 

whence 


T2-/2' 

One  of  the  simplest  forms  of  added  inertia  is  that  of  a  solid  cylin- 
der of  circular  cross  section  vibrating  about  an  axis  through  the 
center  of  the  axis  of  the  cylinder  and  at  right  angles  to  it.  The 
vibrating  mass  may  then  be  in  the  form  of  a  hollow  cylinder  in 
which  the  solid  cylinder  may  be  placed.  If  /  be  the  length  and 
r  the  radius  of  the  solid  cylinder  of  mass  m: 


12 


RIGIDITY.  49 

% 

The  quantities  /  and  r  can  be  obtained  with  sufficient  precision 
by  measurement  with  a  steel  scale  divided  to  mms.,  and  m  may 
be  found  by  a  platform  balance.  In  the  above  formula  for  i,  it 
is  assumed  that  the  axis  of  rotation  is  perpendicular  to  the  axis 
of  the  cylinder.  That  this  may  be  so  the  carrier  must  be  care- 
fully leveled.  This  may  be  done  by  supporting  close  under  it  a 
rod  that  is  carefully  leveled  by  a  spirit-level  and  comparing  the 
carrier  as  it  swings  with  the  leveled  rod.  The  end  of  the  vibrating 
mass  should  be  provided  with  an  index,  such  as  a  vertical  needle. 
A  stationary  vertical  wire  is  placed  in  front  of  this  index  when  the 
latter  is  in  the  position  of  rest.  The  body  is  set  vibrating  through 
an  angle  of  between  60°  and  90°,  all  pendulum  vibrations  being 
carefully  suppressed. 

The  time  of  vibration  may  be  found  by  much  more  accurate 
methods  than  simply  timing  a  certain  number  of  vibrations. 
The  most  common  methods  for  accurately  timing  vibrations  are 
the  "method  of  coincidences"  and  the  "method  of  passages." 
-The  former  is  especially  useful  for  finding  the  time  of  vibration 
of  a  pendulum  whose  half  period  is  approximately  one  second 
(Exp.  VII).  The  method  of  passages  will  be  found  suitable  for 
the  present  experiment.  It  consists  in  noting  as  accurately  as 
possible  the  time  of  every  nth  passage  of  the  vibrating  system 
through  its  mean  position  or  position  of  rest.  This  will  be  spoken 
of  simply  as  a  "passage."  The  value  to  be  chosen  for  n  is  a 
matter  of  convenience  when  two  observers  work  together,  one 
counting  the  seconds  and  the  other  noting  the  passages,  or  when 
a  single  observer  has  a  chronometer  in  front  of  him.  But  a 
single  observer  noting  time  by  a  clock  circuit  and  sounder  should 
choose  for  n  an  odd  number  such  that  n  semi-vibrations  occupy 
a  little  more  than  a  minute.  (It  is  supposed  that  there  is  a  minute 
signal,  such  as  the  omission  of  a  tick;  see  page  23.)  The  proper 
value  for  n  and  the  approximate  (to  within  I  or  2  seconds)  time 
required  for  these  vibrations  may  be  found  by  twice  counting 
the  number  of  half-vibrations  in  three  minutes. 

The  passages  are  observed  as  follows:  After  any  minute  sigrial, 

the  seconds  are  counted   until  a  passage  occurs,   for  example, 

from  left  to  right.     The  second  and  fraction  of  a  second  of  this 

passage  is  recorded.     The  remainder  of  the  minute  is  spent  in 

4 


50  MECHANICS. 

0 

calculating  ahead  the  approximate  second  at  which  the  nth 
passage  will  occur  and  in  getting  ready  to  count  seconds  from  the 
minute  signal  up  to  the  time  of  this  expected  passage,  when  the 
exact  second  and  fraction  of  a  second  at  which  the  passage  occurs 
is  noted.  Since  n  is  an  odd  number  this  passage  will  be  from 
right  to  left.  The  time  of  the  2nth  passage  is  then  calculated 
and  observed  in  the  same  way  and  so  on  to  the  iQwth. 

The  following  suggestion  may  aid  in  counting  the  seconds 
and  estimating  fractions  of  a  second.  The  observer  should  keep 
counting  seconds  (not  necessarily  out  loud)  along  with  the  clock; 
when  the  number  of  the  second  is  of  two  or  more  syllables,  the 
accent  should  be  thrown  on  one  syllable  whose  sound  should 
coincide  with  the  tick;  thus,  eleven,  thirteen,  fourteen,  etc., 
twenty-one,  twenty-two,  etc.  The  passage  will  usually  occur 
somewhere  between  two  .  ticks.  To  estimate  at  what  point 
of  time  between  the  two  seconds  the  passage  takes  place,  the 
indications  of  the  eye  may  be  used  to  reinforce  those  of  the  ear. 
Suppose  A  (in  Fig.  14)  to  be  the  mean  position  of  the  index  on 
the  vibrating  body,  then  if  B  and  C  be  its  positions  at  the  fifth 
and  sixth  ticks,  respectively,  and  if  BA 
be  six-tenths  of  the  distance  BC,  it  is  evident 
that  the  true  time  of  passage  is  5.6  seconds. 
With  practice  the  eye  can  become  very 
expert  in  making  such  judgments,  and,  for 

the  purpose  of  attaining  such  skill,  the  method  should  be  used 
from  the  beginning,  although  at  first  not  much  reliance  can  be 
placed  on  the  judgment. 

Further  assistance  is  obtained  by  arranging  the  readings  in 
columns  as  shown  in  the  following  table,  in  which  UL  to  R" 
refers  to  passages  from  left  to  right  and  "R  to  L"  passages  from 
right  to  left  and  the  proper  value  of  n,  obtained  in  the  way  al- 
ready explained,  is  to  be  substituted. 

—  L  to  R—  —  R  to  L— 

(o)    ........  (ion)  ........  (n)    ........  (nw)  ........ 


(6n) 

(Sn)  ........  (iSri)  ........ 


VISCOSITY.  51 

from  which  the  time  of  vibration  is  calculated  thus  : 

min.      sec.  min.      sec. 

(iow)-(o)    =    ••  (nw)-(n)    =   •• 


(i6w)-(6w) 


Mean  of  low  vibrations  =  •  • 

Final  mean  of  ion  vibrations  =  ----  possible  error  =  ---- 

Final  mean  of  one  vibration     =  •  •  •  •          possible  error  =  •  •  •  • 

To  find  the  possible  error  in  the  value  found  for  n,  first  eliminate 
T  and  i  from  the  equation  given  above  and  express  n  in  terms 
of  the  quantities  observed  L,  /,  71,  /  R,  m.  (r2/4  is  so  small  com- 
pared with  /2/i2  that  the  effect  of  the  possible  error  in  the 
former  may  be  neglected.)  T  and  t  come  in  only  in  the  form 
(T2  —  t2)  and  the  possible  error  in  this  may  be  found  by  methods 
stated  on  page  5. 

Questions. 

1.  To  increase  the  accuracy  of  the  result,  which  quantity  would  have  to  be 
measured  more  closely? 

2.  What  sources  of  error  are  there  other  than  those  referred  to  in  the  text? 


XI.  VISCOSITY. 

References — Elementary:  Duff,  §§196-198;  Ames,  pp.  139,  168;  Kimball,  §245; 
Reed  &  Guthe,  §62;  Watson,  §161. — More  Advanced:  McLean's  Practical 
Physics,  §§340-342;  Poynting  &  Thomson  (Prop,  of  Matter),  Chap.  XVIII; 
Watson  (Pr),  §§55-57- 

A  solid  has  rigidity;  that  is,  it  offers  a  continued  resistance  to 
forces  tending  to  change  its  shape.  A  liquid  has  no  rigidity 
and  offers  no  continued  resistance  to  forces  tending  to  change  its 
shape;  that  is,  the  smallest  force  if  given  time  will  produce  an 
unlimited  change  in  the  shape  of  the  liquid.  But  the  rate  at 
which  a  liquid  changes  its  shape  under  a  given  force  is  not  the 
same  for  all  liquids.  Some  liquids  change  very  slowly  and  are 
called  viscous  liquids,  others  change  rapidly  and  are  called  mobile 
liquids.  The  action  of  both  can  be  stated  in  terms  of  a  property 
called  viscosity. 


52  MECHANICS. 

The  viscosity  of  a  fluid  may  be  defined  as  the  ratio  of  the  shear- 
ing stress  in  the  fluid  to  the  rate  of  shear.  From  this  general 
definition  a  simpler  definition  can  be  readily  deduced  A  shear 
consists  essentially  in  the  sliding  of  layer  over  layer  and  the  shear- 
ing stress  is  the  force  per  unit  area  required  to  produce  the  shear. 
Hence  we  have  the  following  equivalent  definition:  "The  co- 
efficient of  viscosity  is  the  tangential  force  per  unit  of  area  of 
either  of  two  horizontal  planes  at  unit  distance  apart,  one  of 

which  is  fixed  while  the  other 
moves  with  unit  velocity,  the 
space  between  the  two  being 
filled  with  the  liquid."  (Max- 
well.) 

(A)  A  vertical  cylinder,  B, 
(Fig.  15)  of  radius  r\  and  length 
/  is  caused  to  rotate  in  a  co-axial 
cylindrical  vessel  C  of  slightly 
greater  internal  radius,  rZl  the 
space  between  the  two  being 
filled  with  the  liquid. 

Let  the  couple  causing  the  ro- 
tation be  L  and  let  the  velocity 
of  the  surface  of  B  be  V.  The 
couple  L  is  equivalent  to  a  force 
F  applied  tangentially  to  the 
surface  of  B  where  L  =  Fr\. 


FIG.  15. 


The  thickness  of  the  film  is  (r2  —  n)-  To  find  an  approximate 
formula  for  /*,  treat  the"  liquid  film  as  a  plane  sheet  between  plane 
solid  surfaces.  Let  A  be  the  mean  of  the  areas  of  the  two  faces 
of  the  liquid  film  or  2irl(ri  +  rz)/2.  Then  by  definition, 

F/A        L(r2-rO 


v/d 

The  above  formula  is  sufficiently  accurate  if  the  film  be  very 
thin.  A  more  complete  solution  of  the  problem  shows  that  a 
more  accurate  formula  is 


VISCOSITY.  53 

(See  Poynting  and  Thomson's  Properties  of  Matter,  Chapter 
XVIII,  p.  213.) 

C  is  enclosed  in  a  water  bath  to  ensure  constancy  of  tempera- 
ture. B  is  driven  by  a  silk  thread  which  is  wrapped  around  one 
of  the  pulleys  attached  to  B.  The  largest  pulley  is  to  be  used 
for  liquids  of  high  viscosity.  The  diameters  of  the  cylinder 
and  of  the  pulley  used  should  be  measured  several  times  by 
calipers.  The  difference  in  the  diameters  of  the  two  cylinders 
may  be  obtained  by  inserting  a  thin  narrow  steel  wedge,  curved 
on  one  side,  between  the  inner  cylinder  and  the  inside  of  the  outer 
cylinder,  marking  the  point  to  which  it  enters,  and  measuring 
the  thickness  at  this  point  with  micrometer  calipers.  This 
determination  should  be  repeated  several  times. 

The  thread  extends  horizontally  over  a  pulley  P  and  carries 
a  scalepan  in  which  suitable  weights  are  placed.  The  scale 
pan  is  released  exactly  on  a  tick  of  the  clock  and  the  time  re- 
quired for  it  to  reach  the  floor  is  noted,  fifths  of  a  second  being 
estimated  as  closely  as  possible.  From  the  distance  of  descent, 
the  mean  time,  and  the  measurements  of  radii,  Fis  calculated. 
The  observations  should  be  repeated  with  various  weights. 

The  observations  are  then  to  be  plotted  with  velocities  as 
abscissas  and  weights  as  ordinates.  The  result  should  be  a  very 
close  approximation  to  a  straight  line.  This  line  should  cut  the 
axis  of  ordinates  a  small  distance  above  the  origin.  The  inter- 
cept represents  the  weight  required  to  turn  B  with  an  infinitely 
small  velocity,  that  is,  the  weight  required  to  overcome  the  fric- 
tion of  the  bearings  and  of  the  pully  P.  If  this  (supposing  it  to 
be  measurable)  be  subtracted  from  all  the  ordinates,  the  result 
will  be  a  second  straight  line  through  the  origin,  proving  the 
fundamental  law  of  viscosity. 

An  interesting  application  of  this  method  is  to  determine  the 
viscosity  and  temperature  variation  of  the  viscosity  of  an  oil. 
For  this  purpose  the  bath  is  heated  to  the  required  temperature 
by  a  Bunsen  burner.  For  greater  constancy  of  temperature  the 
bath  should  be  wrapped  with  asbestos.  Before  the  time  of 
descent  is  observed  the  burner  should  be  removed  and  the  bath 
should  be  thoroughly  stirred.  The  temperature  must  be  main- 
tained constant  until  the  time  of  descent  has  been  determined  for 


54  MECHANICS. 

a  sufficient  number  of  weights  for  construction  of  the  above  plot 
for  the  temperature.  A  mean  value  of  rj  for  a  particular  tem- 
perature may  readily  be  calculated  from  the  slope  of  the  line  and 
the  radii.  Observations  should  be  made  at  several  temperatures 
and  a  curve  plotted  with  temperature  as  abscissas  and  values  of 

rj  as  ordinates. 

Questions. 

1.  What  somewhat  inaccurate  assumptions  are  made  in  deriving  the  first 
formula? 

2.  What  differences  are  there  between  viscous  and  f  rictional  forces  as  respect 
(a)  their  nature  (b)  their  effect  on  the  resultant  motion? 

(B)  The  flow  of  liquid  through  a  capillary  tube  is  essentially 
of  the  nature  of  sliding  of  layer  over  layer.  The  cylindrical  layer 
in  immediate  contact  with  the  tube  remains  fixed  or  at  least 
has  no  motion  parallel  to  the  axis  of  the  tube,  and  the  immedi- 
ately adjacent  layer  slides  over  it,  the  next  layer  slides  over  the 
second,  and  so  on  up  to  the  center  of  the  tube.  (In  a  tube  of 
greater  than  capillary  bore  this  is  not  so,  for  there  are  eddies 
in  the  motion.  This  distinction  is  in  fact  the  best  definition  of 
the  term  capillary.) 

Thus,  if  we  measure  the  force  causing  flow  through  the  tube 
and  the  rate  of  flow,  we  shall  be  in  a  position  to  deduce  the  co- 
efficient of  viscosity  of  the  fluid.  In  fact,  if  M  be  the  mass  of 
liquid  of  density  d  that  flows  in  time  /,  through  a  vertical  tube  of 
length  /  and  radius  of  bore  r,  and  if  h  be  the  vertical  distance  from 
the  level  of  the  liquid  in  the  reservoir  above  the  tube  to  the  lower 
end  of  the  tube,  the  coefficient  of  viscosity  is 


Suppose  all  the  liquid  in  a  capillary  tube  of  length  /  and  radius  r  to  be 
solidified  except  a  tubular  layer  of  mean  radius  x  and  thickness  dx.  If  there 
be  a  difference  of  pressure  p  (per  unit  of  area)  between  the  two  ends,  the  solid 
will  attain  a  steady  velocity  such  that  the  viscous  resistance  just  equals  the 
whole  difference  of  pressure  on  its  ends.  Hence  it  follows  from  the  definition 
of  the  coefficient  of  viscosity  that 

2-n-xlvr]  pxdx 

-te-=**p.     Hence,  v  =  -^. 

If  q  be  the  volume  of  the  core  that  flows  out  per  second, 

pirx3dx 


VISCOSITY.  55 

Suppose  now  another  layer  liquefied.  There  will  follow  a  further  flow  repre- 
sented by  the  same  expression  but  with  a  different  value  for  x.  Let  the  process 
be  continued  until  the  whole  is  liquid,  then  the  whole  flow  per  second,  0,  will 
be  the  sum  of  all  the  values  of  q  for  values  of  x  between  o  and  r.  Hence 


- 

-  8/77* 

If  the  tube  be  vertical  and  the  flow  be  due  to  gravity  only,  instead  of  p  we 
must  put  gdh.  If  M  be  the  mass  of  density  d  that  flows  out  in  time  t, 

vgdWt 
M=Qdt  -Mi  =  -8^p 

In  the  above  it  was  tacitly  assumed  that  the  liquid  adheres  to  the  tube  without 
any  slip.  If  there  were  any  slip  the  outflow  would  be  increased  by  it  and  the 
above  expresssion  would  not  hold.  Poiseuille  and  others  verified  the  above 
formula  in  all  cases,  thus  showing  that  no  slip  occurs.  (A  more  formal  proof 
of  the  above  equation  is  given  in  Tait's  "Properties  of  Matter,"  §317.) 

A  piece  of  capillary  tubing  should  be  chosen  whose  bore  is 
as  n'early  as  possible  circular  in  section.  This  can  be  tested  by 
examining  the  ends  under  a  micrometer  microscope  (see  p.  13). 
If  the  section  is  found  to  be  nearly  circular  the  principal  di- 
ameters of  the  bore  should  be  measured.  This  should,  however, 
only  be  regarded  as  a  preliminary  measurement,  serving  as  a  test 
of  the  circularity  of  the  bore  and  a  check  on  the  following  more 
satisfactory  method. 

The  mean  radius  of  the  bore  can  be  best  determined  by  weigh- 
ing the  amount  of  mercury  that  fills  a  measured  length  of  the 
tube.  For  this  purpose  the  tube  should  be  first  cleaned  by  attach- 
ing it  to  the  end  of  a  rubber  tube,  at  the  other  end  of  which  is  a 
hollow  rubber  ball,  and  thus  drawing  through  it  and  forcing  out 
a  number  of  times  (i)  chromic  acid;  (2)  distilled  water;  (3)  alco- 
hol, and  finally  drying  it  by  sucking  air  through  it.  Then  draw 
into  the  tube  a  column  of  clean  mercury  and  measure  its  length 
several  times  by  a  comparator  (see  p.  14). 

The  mass  of  the  mercury  should  next  be  ascertained  by  weigh- 
ing it  with  great  care  in  a  sensitive  balance  (for  full  directions 
see  pp.  20-23).  The  mercury  should  not  be  dropped  directly 
on  the  scale  pan,  but  into^a  watch-glass  or  paper  box  placed  on  the 
scale  pan.  From  these  measurements  and  the  density  of  mercury 
at  the  temperature  of  observation  (see  Table  VII)  the  diameter 
of  the  bore  is  obtained.  It  may  be  noted  that  since  it  is  r4  that 
is  used  in  the  formula  for  viscosity  and  r2  that  is  obtained  directly 


MECHANICS. 


from  the  mercury  measurements  of  the  bore  the  value  of  r  need 
not  be  deduced.  The  length  of  the  tube  may  be  measured  by  the 
comparator  in  the  same  manner  as  the  thread  of  mercury. 
.  The  tube  is  then  attached  vertically  by  a  rubber  connection 
to  a  funnel  and  the  mass  of  water  that  flows  through  the  tube  in  a 
given  time  found  by  weighing  a  beaker  (i)  empty  and  (2)  con- 
taining the  water  that  has  passed.  The  time  is  obtained  by 
observing  a  clock  ticking  seconds  or  a  chronometer.  It  is  evi- 
dent that  the  greater  the  whole  time  the  less  the  percentage  error 
in  time  due  to  errors  in  observing  the  time  of  starting  and  stop- 

^ 7  ping,  and  so,  too,  the  greater  the  whole  mass 

the  less  the  percentage  error  in  weight.  Hence 
the  time  and  the  mass  should  be  sufficiently 
great  to  make  the  percentage  errors  in  them  less 
than  those  in  /  and  r*.  To  prevent  evaporation 
from  the  beaker  it  should  be  covered  by  a  sheet 
of  paper  pierced  by  a  hole  through  which  the 
tube  passes.  While  the  liquid  is  flowing  the 
temperature  of  the  water  in  the  funnel  should  be 
noted. 

The  value  of  h  is  the  mean  of  its  values  at  the 
beginning  and  end  of  the  flow.  These  values  are 
best  obtained  by  a  cathetometer  (p.  17).  For 
this  purpose  a  very  simple  form  of  instrument 
may  be  used.  A  vertical  scale  is  placed  near 
the  apparatus  for  viscosity  and  the  cathetometer 
(a  telescope  that  may  be  leveled,  movable  along  a 
vertical  column  that  may  be  made  truly  vertical) 
placed  so  that  its  telescope  (leveled  to  horizontality)  may  be 
turned,  so  that  the  intersection  of  its  cross-hairs  coincides  alter- 
nately with  the  image  of  the  water  surface  and  that  of  the  scale. 
This  gives  the  level  of  the  surface  of  the  liquid  on  the  vertical 
scale.  The  level  of  the  lower  end  of  the  vertical  tube  is  obtained 
in  the  same  way,  whence  h  is  obtained. 

The  viscosity  of  alcohol  may  be  measured  by  the  same  means, 
particular  care  being  taken  to  prevent  evaporation.  The  possi- 
ble error  of  the  result  is  readily  calculated  from  the  possible  errors 
of  the  separate  measurements.  The  possible  error  of  r  is  not 


,_i  

TT 

T> 
] 

h 
.J 

I 

. 

FIG.  16. 


SURFACE   TENSION.  57 

needed,  but  that  of  r4  should  be  deduced  directly  from  the  de- 
termination of  r2. 

Questions. 

1.  Could  the  radius  be  found  satisfactorily  by  measurements  with  a  micro- 
meter microscope?     Explain. 

2.  What  mass  of  the  liquid  tested  would  flow  per  minute  through  a  tube 
i  mm.  in  diameter  and  I  meter  long,  under  a  constant  head  of  2  meters? 

3.  Two  square  flat  plates  of  20  cm.  edge  are  separated  by  i  mm.  of  this  liquid. 
What  force  would  be  required  to  move  one  with  a  velocity  of  30  cm.  per  second, 
the  other  being  at  rest? 


XII.  SURFACE  TENSION. 

References — Elementary:  Duff,  §§206-214;  Ames,  pp.  182-190;  Crew,  §§149-160; 
Kimball,  §§255-264;  Reed  &  Guthe,  §§83-90;  Spinney,  §§144-150;  Watson, 
§§155-160. — More  Advanced:  Gray's  Treatise  on  Physics,  I,  §§683-696; 
McLean's  Practical  Physics,  §§132-139;  Poynting  &  Thomson,  (Prop,  of 
Matter),  Chap.  XIV. 

The  height  to  which  liquid  rises  or  is  depressed  in  a  capillary 
tube  depends  on  the  surface  tension  of  the  liquid,  the  angle  of 
capillarity,  and  the  radius  of  the  tube.  From  measurements  of 
the  height,  h,  and  radius,  r,  the  surface  tension  is  deduced  if 
the  angle  of  capillarity  is  known,  for  (see  references) 

T=    rdgh 
2  cos  a 

d  being  the  density  of  the  liquid  and  g  the  acceleration  of  gravity. 
In  the  case  of  perfectly  pure  distilled  water  the  angle  of  capillary 
a  or  the  angle  at  which  the  surface  of  the  liquid  meets  the  glass, 
is  zero  and  so  cos  a  =  I. 

It  is  important  that  the  capillary  tube  be  quite  clean.  The 
cleaning  should  be  performed  with  chromic  acid  and  distilled 
water.  The  height  of  the  liquid  in  the  tube  can  be  measured  in 
two  ways.  One  method  is  to  place  a  scale  etched  on  mirror 
glass  behind  the  tube.  The  mean  level  of  the  meniscus-shaped 
surface  of  the  liquid  in  the  tube  and  the  ordinary  plane  surface 
of  the  liquid  in  the  vessel  should  be  read.  A  preferable  method 
is  to  measure  the  distance  between  the  two  surfaces  with  a  cathe- 
tometer  (see  p.  17).  The  apparatus  should  be  surrounded  by  a 
water  bath  to  maintain  a  constant  temperature. 


58  MECHANICS. 

To  make  certain  that  the  inner  surface  of  the  tube  is  wet  by  the 
liquid  and  that  the  angle  of  capillarity  is  zero,  the  tube  should  be 
thrust  deeper  into  the  liquid  and  then  withdrawn  before  the 
levels  of  the  surfaces  are  read.  Or,  if  the  liquid  is  contained  in 
a  side  neck  test  tube  and  the  capillary  passes  through  a  rubber 
stopper  and  dips  in  the  liquid,  the  liquid  may  temporarily  be 
forced  up  the  capillary  by  compressing  slightly  the  air  above  the 
liquid  outside  the  capillary  by  blowing  air  into  the  side  tube. 
This  should  be  repeated  and  the  height  read  several  times. 
If  the  motion  of  the  liquid  in  the  tube  is  sluggish  or  uncertain, 
the  tube  should  be  more  carefully  cleaned.  Finally,  the  mean 
point  to  which  the  liquid  rises  in  the  tube  should  be  marked 
on  the  tube  by  a  sharp  file. 

The  tube  should  then  be  carefully  broken  at  the  point  marked 
and  its  diameter  should  be  carefully  measured  by  means  of  a 
micrometer  microscope  (see  p.  13).  If  the  section  of  the  bore  is 
not  circular,  the  greatest  and  least  diameters  should  be  carefully 
measured  and  the  mean  taken,  but  if  they  differ  very  much  the 
result  will  not  be  satisfactory. 

The  whole  should  be  repeated  with  as  many  tubes  of  different 
sizes  as  time  will  permit.  The  temperature  at  which  the  work  is 
performed  should  be  stated. 

If  time  permit,  determine  also  the  surface  tension  at  another 
temperature  and  from  the  two  observations  calculate  the  tempera- 
ture coefficient  of  the  surface  tension. 

Make  an  estimate  of  the  possible  error  for  the  results  obtained 
by  one  of  the  tubes. 

Questions. 

1.  Are  the  errors  of  measurement  sufficient  to  explain  the  differences  be- 
tween results  with  different  tubes? 

2.  What  other  sources  of  error  may  there  be? 

3.  How  could  the  surface  tension  of  mercury  be  obtained  in  an  analogous 
way? 

4.  How  high  would  the  liquid  tested  rise  in  a  tube  o.i  mm.  in  diameter? 


HEAT. 


25.  Radiation  Correction  in  Calorimetry. 

References — Ostwald's  Phys.  Chem.  Meas.,  p.  124-127;  Poyntingand  Thomson, 

Heat,  Chap.  XVI. 

A  body  which  is  above  the  temperature  of  surrounding  bodies 
falls  in  temperature  at  a  rate  that  is  proportional  to  the  excess 
of  its  temperature  above  that  of  its  surroundings.  This  is  New- 
ton's Law  of  Cooling.  If  the  mean  excess  of  the  body's  tempera- 
ture in  any  time  be  known  and  also  its  rate  of  loss  of  temperature 


FIG.  17. 

at  some  particular  excess,  its  mean  rate  of  loss  of  temperature 
is  readily  deduced,  and  this  multiplied  by  the  time  for  which  the 
mean  is  taken  will  give  the  whole  loss  of  temperature. 

Consider  the  case  of  the  heating  of  a  vessel  containing  water 
by  the  passage  of  steam  into  the  water.  If  a  curve  (Fig.  17 — 
from  o  upwards)  showing  the  rise  of  temperature  of  the  water 

59 


60  HEAT. 

be  drawn  and  the  same  continued  after  the  water  has  reached  its 
highest  temperature,  the  latter,  or  straight  line  part  of  the  curve, 
will  give  the  rate  of  loss  of  temperature  at  the  highest  temperature 
attained.  Let  us  denote  this  rate  by  r  (degrees  per  minute). 
If  the  excess  of  temperature  when  the  temperature  is  highest  is  / 
and  if  the  mean  excess  during  the  whole  time  of  rise  of  tempera- 
ture is  /'  then  the  mean  rate  of  cooling  was  by  Newton's  Law 
rt'./t  and  this  multiplied  by  the  whole  time  of  rise  of  temperature, 
T  (minutes),  gives  the  whole  loss  of  temperature.  Hence  the 
final  (highest)  temperature  must  be  corrected  by  addition  of 
rTt' '//.  If  the  curve  of  rise  of  temperature  is  a  straight  line,  /' 
is  half  of  /  and  the  correction  is  rT/2.  When  the  curve  is  not 
approximately  a  straight  line  the  whole  time  T  must  be  divided 
into  a  number  of  intervals  (each  perhaps  of  30  sec.)  and  /'  must 
be  obtained  by  averaging  the  excesses  in  these  intervals. 

When  the  calorimeter  is  cooled  below  the  temperature  of  the 
room  (e.  g.,  by  adding  ice  to  the  water)  the  calorimeter  gains 
temperature  by  radiation  from  the  surroundings;  but  the  above 
method  will  still  apply  except  that  we  shall  have  to  do  with  rates 
of  warming  instead  of  rates  of  cooling  and  the  correction  of  the 
final  temperature  will  be  subtractive. 

If  the  calorimeter  starts  below  the  temperature  of  the  room 
and  is  heated  above  it,  the  correction  must  be  made  in  two  parts 
as  above.  In  this  case  we  must  find  the  initial  rate  of  warming 
(before  the  hot  body  is  placed  in  the  calorimeter)  and  also  the 
final  rate  of  cooling  (after  the  highest  temperature  was  attained). 
The  correction  will  also  be  in  two  parts  when  the  calorimeter 
starts  above  the  room  temperature  and  ends  below  it. 

If  the  main  rise  of  temperature  is  closely  represented  by  a 
straight  line,  it  is  easily  shown  that  the  correction  amounts  to 
the  algebraic  average  of  the  initial  and  final  rates  multiplied  by 
the  whole  time  that  the  calorimeter  is  heating  or  cooling.  In  fact, 
if  the  water  is  T\  minutes  below  the  temperature  of  the  room  and 
T%  minutes  above  the  room  temperature,  the  radiation  correc- 
tion is  (r2r2-7>i)/2  and  this  differs  from  (Ti  +  T2)  (n  +  r2)/2 
by  (2V2— 2VO/2,  which  is  zero,  since  under  these  circumstances 
the  rate  is  proportional  to  the  time  that  the  water  is  above  or 
below  the  room  temperature.  This  method  is  particularly  useful 


THE    BECKMANN    THERMOMETER.  6 1 

in  cases  where  the  surrounding  temperature  is  indefinite  (Exp. 
XXVII,  for  example). 

In  every  calorimetry  experiment  where  the  temperature 
changes,  this  radiation  correction  must  be  applied,  and  therefore 
the  initial  and  final  rates  of  change  of  temperature  must  be  de- 
termined. The  rate  may  usually  be  found  with  sufficient  ac- 
curacy by  reading  the  temperature  every  minute  for  five  minutes. 
In  very  accurate  work,  more  careful  methods  must  be  applied. 

26.  The  Beckmann  Thermometer. 

References — Watson's    Practical    Physics,  §102;    Findlay,   Practical   Physical 
Chemistry,  pp.  114-117;  Ostwald  Phys.  Chem.  Meas.,  pp.  119-120. 

The  Beckmann  thermometer  is  used  for  determining  changes 
in  temperature.  The  bulb  is  large  and  the  stem  is  small  so  that  a 
small  change  of  temperature  is  shown  by  a  large  change  in  read- 
ing. The  amount  of  mercury  may  be  varied,  and  the  tempera- 
ture corresponding  to  a  particular  reading  will  vary  with  the 
amount  of  mercury  in  the  bulb  and  stem.  There  is  a  reservoir 
at  the  end  of  the  stem  into  which  surplus  mercury  may  be  driven 
by  warming  the  bulb.  A  gentle  jar  will  detach  the  mercury  in 
this  reservoir  when  sufficient  has  been  expelled.  If  one  desires 
to  study  high  temperature  changes,  the  bulb  is  warmed  until 
the  thread  of  mercury  extends  to  the  reservoir,  when  the  mercury 
in  the  reservoir  is  joined  to  it.  The  bulb  is  then  allowed  to  cool 
until  sufficient  mercury  has  been  drawn  over,  when  the  thread  is 
detached  from  the  mercury  in  the  reservoir  by  a  gentle  jar. 
Several  trials  are  often  necessary  before  the  proper  amount  of 
mercury  is  secured.  In  an  improved  type  of  Beckmann  ther- 
mometer, two  reservoirs  are  provided,  one  of  which  has  a  scale 
which  tells  the  amount  of  mercury  required  in  that  reservoir  for 
different  ranges  of  temperature. 

Beckmann  thermometers  are  delicate  and  expensive  and  must 
be  handled  with  the  greatest  care. 


62  HEAT. 

XIII.  THERMOMETER  TESTING. 

References — Elementary:  Duff,  §267;  Ames,  pp.  220-224;  Crew,  §§242-247; 
Edser  (Heat],  Chap.  II;  Kimball,  §§366-370;  Reed  &  Guthe,  §§149-152; 
Spinney,  §§154-156;  Watson,  §§177-182. — More  Advanced:  Poynting  & 
Thomson  (Heat},  pp.  3-8;  Watson,  (Pr.),  §§59-69. 

The  readings  of  a  thermometer  gradually  change  for  a  long 
time  after  the  thermometer  has  been  filled.  The  cause  of  this 
is  the  gradual  recovery  of  the  bulb  from  the  effect  of  the  very 
great  heating  to  which  the  glass  was  subjected  when  the  ther- 
mometer was  made.  The  shrinkage  is  rapid  at  first  and  slower 
afterward,  but  many  continue  for  years.  Hence  the  necessity  for 
re-determining,  from  time  to  time,  the  so-called  "fixed  points" 
of  a  thermometer,  namely,  the  reading  in  melting  ice,  and  that 
in  steam  at  standard  pressure.  When  the  thermometer  is  first 
graduated  it  is  usually  done  by  determining  the  fixed  points  and 
dividing  the  distance  between  them  into  100  equal  parts  laid  off 
on  the  stem.  This  assumes  that  the  bore  is  uniform  or  that,  by 
calibration  of  the  bore,  the  variations  of  the  bore  are  determined 
and  allowed  for  in  a  table  of  corrections  to  be  applied  to  the  read- 
ings of  the  thermometer  in  order  to  obtain  the  true  temperature. 
Usually  the  variations  of  the  bore  are  too  small  to  have  any  ap- 
preciable effect  except  in  cases  where  extreme  accuracy  is  aimed 
at.  Nevertheless,  every  thermometer  needs  to  be  carefully 
examined  in  this  regard.  Let  us  suppose  that  on  the  scale  laid 
off  on  the  stem  the  true  readings  in  ice  and  steam  have  been  ob- 
tained and  for  the  moment  let  us  suppose  that  the  bore  is  quite 
uniform.  To  see  how  to  make  corrections  for  other  points  on 
the  scale  we  must  consider  the  elementary  definition  of  tempera- 
ture. 

Temperature  on  the  mercury  scale  is  defined  by  the  expansion 
of  mercury  (relatively  to  glass).  Let  Vm  be  the  volume  of  a  mass 
of  mercury  at  the  temperature  of  steam  under  a  pressure  of  76  cm. 
and  let  VQ  be  its  volume  at  the  temperature  of  melting  ice.  The 
degree  is  defined  as  the  rise  of  temperature  that  would  produce 
an  expansion  of  (^100  —  flo)/ioo,  and  T°  above  zero  is,  therefore, 
the  rise  of  temperature  that  will  produce  an  expansion  of 
T(VIQO  —  v0)/ioo.  Hence  if  at  T°  the  volume  of  the  mercury  be  v, 


THERMOMETER   TESTING.  63 


IOO 


This  definition  depends  only  on  the  expansion  of  mercury  and 
the  expansion  of  the  particular  glass  used  and  is  otherwise  inde- 
pendent of  the  size  and  shape  of  the  thermometer.  Now  regard 
the  thermometer  tube  under  test  as  simply  a  graduated  cylinder 
of  constant  cross  section  containing  mercury.  Let  the  height 
of  the  mercury  as  read  on  the  scale  when  the  thermometer^  is  in 
melting  ice  be  a;  when  it  is  a  steam  at  76  cm.  let  it  be  b,  and 
when  it  is  at  the  temperature  T,  let  it  be  t.  Then 

v  —  VQ      t  —  a 


^100  ~  VQ     b  —  a 
Hence 


where  T  is  the  true  temperature  when  the  reading  of  the  ther- 
mometer is  /.  By  this  equation  values  of  T  for  values  of  t  for 
every  five  degrees  should  be  calculated.  Having  thus  drawn 
up  a  table  of  true  temperatures  we  subtract  the  scale-reading 
from  the  true  temperature  and  thus  get  a  correction  (positive 
or  negative),  which  added  to  the  scale-reading  gives  the  true 
temperature.  These  corrections  should  also  be  tabulated. 

This  is  on  the  assumption  that  the  bore  is  sensibly  uniform. 
The  only  quite  satisfactory  method  of  testing  this  is  to  calibrate 
the  bore  by  measuring  the  length  of  a  short  thread  of  mercury 
at  different  positions  in  the  tube.  This  process  requires  consid- 
erable time  and  the  following  will  usually  suffice:  Two  ther- 
mometers for  which  tables  of  true  readings  have  been  drawn 
up  as  above,  are  compared  at  regular  intervals  (say  every  five 
degrees)  between  zero  and  100°  by  being  used  simultaneously 
to  measure  the  temperature  of  a  body.  If,  after  corrections,  the 
readings  of  the  thermometers  are  not  sensibly  different,  this 


64  HEAT. 

shows  that  the  bores  of  both  must  be  practically  uniform. 
If  they  do  differ  appreciably,  then  the  bore  of  one  or  both  must 
be  variable.  If  they  be  compared  with  a  third  thermometer, 
the  one  with  the  variable  bore  will  be  detected  and  it  must  be 
then  calibrated. 

Testing  Zero-point. — A  calorimeter  consisting  of  a  small  copper 
vessel  inside  of  a  larger  is  suitable  for  holding  the  ice.  Both 
vessels  should  be  washed  in  ordinary  tap  water.  The  space  be- 
tween the  two  vessels  should  be  filled  with  cracked  ice,  and  the 
inner  vessel  filled  with  cracked  ice  and  then  distilled  water  poured 
in  until  the  vessel  is  filled  to  the  brim.  The  thermometer  having 
been  washed  clean,  is  inserted  in  the  inner  vessel,  just  sufficient 
of  the  stem  being  exposed  to  admit  of  the  zero  being  observed. 
When  the  reading  has  fallen  to  i°  the  reading  should  be  observed 
every  minute  until  it  is  stationary  for  five  minutes.  This  sta- 
tionary temperature,  read  to  o.i  degree,  is  the  true  zero  point, 
or  a  in  the  above  equation. 

Sources  of  Error. 

(1)  Impurity  in  the  ice  or  water. 

(2)  The  presence  of  water  above  o°  near  the  bulb  of  the  ther- 
mometer. 

Testing  Boiling-point. — The  form  of  boiler  used  for  this  test 
consists  of  a  vessel  for  boiling  water  surmounted  by  a  tube  up 
which  the  steam  passes,  this  tube  being  enclosed  in  another  down 
which  the  steam  passes  to  an  exit  tube  and  a  pressure  gauge 
(see  Fig.  18).  Half  fill  the  lower  part  of  the  vessel  with  water. 
As  a  preliminary  experiment,  push  the  thermometer  to  be 
tested  through  a  cork  in  the  top  Until  the  bulb  alone  is  in  the 
steam. 

Apply  heat,  adjusting  it  carefully  as  boiling  begins,  so  that  the 
pressure  inside,  as  indicated  by  the  pressure  gauge,  shall  not 
materially  exceed  atmospheric  pressure.  Some  excess  is,  of 
course  necessary,  if  there  is  to  be  a  free  flow  of  steam.  What 
excess  is  permissible  may  be  deduced  from  the  consideration  that 
a  rise  of  pressure  of  I  cm.  (of  mercury  column)  corresponds  to  a 
rise  of  boiling-point  of  0.373°  (see  Table  XIII).  If  water  is 
used  in  the  pressure  gauge,  a  pressure  of  I  cm.  of  water  column 
would  correspond  to  only  0.03°  rise  of  steam  temperature.  If 


THERMOMETER   TESTING. 


L 


the  thermometer  be  graduated  to  degrees  only,  an  error  of  0.03° 
in  finding  the  boiling-point  is  negligible. 

After  several  careful  readings  of  the  thermometer,  repeat  with 
the  thermometer  lowered  until  only  the  boiling  point  is  visible. 
The  bulb  must  not,  however,  reach  the  water.  The  mean  difference 
in  readings  gives  the  stem  correction,  which  must  be  applied 
when  the  stem  cannot  be  immersed  (e.  g.,  Exp.  LIV,  LXVIII). 

Read  the  barometer  and  reduce  the  height  to  zero  degrees 
(p.  19).  To  the  boiling-point  found 
a  correction  must  be  applied  for  the 
difference  between  the  atmospheric 
pressure  at  the  time  and  that  of  a 
standard  atmosphere  (76  cm.  of 
mercury).  Find  from  Table  XIV 
the  true  temperature  of  the  steam 
at  this  pressure,  and  the  difference 
between  the  boiling-point  observed 
on  the  thermometer  and  this  tem- 
perature. Since  this  temperature 
is  always  within  a  few  degrees  of 
1 00°,  the  thermometer  will  have 
practically  the  same  error  at  100°. 
Therefore  b  in  the  above  equation 
may  be  taken  as  100  plus  or  minus 
the  difference  between  the  last  ob- 
served boiling-point  and  the  true 
boiling  temperature. 

Comparison  of  Two  Thermometers. 
—The  most  satisfactory  method  is 
to  immerse  the  thermometers  in 
steam  above  water  boiling  under  a  pressure  that  can  be  regu- 
lated. A  simple  means  that  is  sufficient  if  the  thermometers  are 
of  the  same  length  and  graduated  to  degrees  only,  is  to  use  the 
thermometers  simultaneously  to  find  the  temperature  of  a  block 
of  good  conducting  material  (copper  or  brass)  immersed  in  a 
vessel  of  water  the  temperature  of  which  can  be  gradually  raised 
by  a  burner.  The  thermometers  should  be  thrust  in  holes  close 
together  in  the  block  and  before  each  reading  the  burner  should 
5 


FIG.  18. 


66  HEAT. 

be  removed  and  the  water  well  stirred  for  a  minute  so  that  the 
temperature  of  the  block  shall  become  uniform. 

Questions. 

1.  Which  should  be  determined  first,  boiling-point  or  freezing-point,  and 
why? 

2.  Calculate  (f?om  the  coefficient  of  expansion  of  mercury)  the  error  due 
to  non-immersion  of  the  stem. 

3.  Why  is  there  no  need  to  take  account  of  barometric  pressure  in  finding 
the  zero-point? 

4.  Could  a  salt  solution  be  used  in  the  boiling  point  apparatus?     Explain. 

XIV.    TEMPERATURE  COEFFICIENT  OF  EXPANSION. 

References — Elementary:  Duff,  §§274-276;  Ames,  pp.  229-233;  Crew,  §§263-265; 
Edser  (Heat),  Chap.  \\l\Kimball,  §§379-382;  Reed  &  Guthe,  §154;  Spinney, 
§§158-160;  Watson,  §§184-5. — More  Advanced:  Poynting  &  Thomson 
(Heat),  Chap.  II. 

For  measuring  the  thermal  expansion  of  a  body,  choice  may 
usually  be  made  from  a  variety  of  methods.  The  particular 
method  chosen  will  depend  on  the  form  of  the  specimen.  The 
expansion  of  a  metal  rod  may  be  measured  by  means  of  a  sphero- 
meter  or  by  means  of  two  reading  microscopes  focused  on  definite 
marks  near  the  ends  of  the  specimen.  The  expansion  of  a  wire 
is  best  measured  by  an  optical  lever  method.  The  expansion  of  a 
solid  of  irregular  form  can  be  found  by  a  hydrostatic  method, 
namely,  by  weighing  it  in  a  liquid  at  different  temperatures, 
it  being  supposed  that  the  density  of  the  liquid  at  different 
temperatures  is  known. 

(A)  Expansion  of  a  Metal  Rod. — The  rod  is  supported  at  the 
lower  end  on  a  firm  point  and  is  heated  by  being  enclosed  in  a 
tube  through  which  steam  is  passed  from  a  boiler.  A  sphero- 
meter  (see  p.  14)  is  so  supported  that  the  end  of  the  screw  can 
be  brought  down  on  the  flat  end  of  the  rod.  The  spherometer 
is  supported  in  the  hole,  slot,  and  plane  method,  so  that  its  posi- 
tion is  definite  and  not  liable  to  be  disturbed  by  thermal  expan- 
sion of  the  supporting  surface. 

Greater  accuracy  is  obtained  if  the  rod  is  entirely  inside  the 
steam  jacket.  This  requires  a  collar  which  slips  over  the  end 
of  the  rod  and  extends  outside.  Both  collar  and  lower  point 
are  preferably  of  "invar."* 

*  A  non-expansive,  nickel-steel  alloy,  obtainable  from  Agar  Baugh,  London. 


TEMPERATURE   COEFFICIENT   OF   EXPANSION. 


The  rod  is  first  measured  by  means  of  an  ordinary  meter  scale 
divided  to  mms.  It  is  then  accurately  placed  in  position  in  the 
heating  tube,  the  end  of  the  rod  projecting  through  corks. 
Through  the  cork  at  the  upper  end  should  also  pass  a  glass  tube 
for  the  entry  of  the  steam,  while  a  similar  tube  at  the  lower  end 
serves  to  drain  off  the  water. 

At  least  six  readings  of  the  spherometer  scales  should  be  made 
at  the  room  temperature.  Then  pass  steam 
into  the  jacket  about  the  rod.  Every  few 
minutes  read  the  temperature  of  the  interior  as 
given  by  two  thermometers  at  different  heights 
and  read  the  spherometer.  When  the  tempera- 
ture has  become  constant,  make  at  least  six 
readings  of  the  spherometer  and  several  readings 
of  the  thermometer.  Always  estimate  tenths  of 
the  smallest  division.  From  the  difference  in 
spherometer  readings,  the  length,  and  the  change 
in  temperature,  calculate  the  coefficient  of  expan- 
sion. 

(B)  Expansion  of  a  Wire.  —  For  this  an  optical 
lever  method  is  most  suitable.  A  mechanical 
lever  or  system  of  levers  is  sometimes  employed 
for  magnifying  small  motions.  A  ray  of  light 
reflected  from  a  mirror  that  is  tilted  by  the  ex- 
pansion serves  the  purpose  of  a  long  index  arm 
much  better,  inasmuch  as  it  has  no  weight  itself 
and  may  be  taken  as  long  as  we  wish.  The 
wire  is  hung  vertically,  the  lower  end  being 
solidly  clamped,  and  the  upper  end  carrying  a 
sleeve  on  which  rests  one  leg  of  a  small  three- 
legged  bench,  on  which  a  mirror  is  mounted. 


FIG.  19. 


Both  clamp  and  sleeve  are  preferably  of  invar.  The  other  two  legs 
rest  on  a  fixed  bracket.  The  wire  is  enclosed  by  a  tube  through 
which  a  current  of  steam  is  passed  from  a  boiler  and  into  which 
two  thermometers  are  thrust  to  read  the  temperature.  A  drainage 
tube  at  the  lower  end  allows  the  escape  of  water.  The  wire  is 
prolonged  above  the  mirror  and  is  attached  to  a  spring  by  which 
the  wire  is  kept  stretched.  The  image  in  the  mirror  of  a  vertical 


68  HEAT. 

scale  is  observed  by  a  reading  telescope  (see  p.  23  for  adjust- 
ments), and  the  change  of  reading,  d,  on  the  horizontal  cross- 
hairs of  the  telescope,  produced  by  the  expansion  is  noted.  Let 

the  length  of  the  wire  between  the 

T     clamp  and  the  support  of  the  lever 

J*  be  /,  and  let  the  distance  between 
the  point  of  the  movable  leg  and 
the  line  of  the  other  lees  be  a.  Let 

r  IG.  2O. 

the  distance  of  the  scale  from  the 
mirror  be  L,  and  the  change  of  temperatures  be  (t^  —  t\).  Then, 
remembering  that  a  ray  of  light  reflected  from  a  mirror  turns 
through  twice  the  angle  that  the  mirror  turns  through,  it  is  easily 
seen  from  the  figure  that  the  expansion  is  ad/2L  and  the  coeffi- 
cient of  expansion  is 

ad 
e  = 


The  most  difficult  quantity  to  determine  with  a  high  degree 
of  precision  is  a.  It  may  be  measured  by  means  of  a  micrometer 
microscope  or  a  dividing  engine  (see  p.  16).  A  simpler  and  more 
accurate  method  is  to  place  the  optical  lever  so  that  the  movable 
leg  is  on  the  (vertical)  screw  of  a  micrometer  caliper  (p.  13), 
while  the  other  legs  are  on  a  fixed  support  and  then  focus  the 
telescope  and  scale  on  the  mirror.  When  the  screw  is  turned  the 
movable  leg  is  raised  a  known  amount.  Repeat  this  calibration 
several  times  and  calculate  a  from  the  mean.  From  a,  the  dis- 
tance between  the  mirror  and  the  scale,  and  the  scale  readings, 
e  is  deduced. 

This  calibration  may  be  avoided  by  placing  two  legs  of  the 
mirror  bench  upon  the  collar  attached  to  the  wire  and  resting 
the  third  leg  directly  upon  the  micrometer  screw.  The  extension 
may  also  be  measured  by  Searle's  combination  of  level  and  micro- 
meter scerw  (see  end  of  Exp.  IX). 


Questions. 

1.  What  sources  of  error  remain  uncorrected? 

2.  What  other  methods  are  there  for  determining  the  coefficient  of  linear 
expansion? 


COEFFICIENT   OF   APPARENT    EXPANSION   OF   A   LIQUID.         69 

XV.    COEFFICIENT  OF  APPARENT  EXPANSION  OF  A 

LIQUID. 

References — Elementary:  Duff,  §§277-278;  Ames,  pp.  233-235;  Crew,  §§266- 
267;  Edser  (Heat),  pp.  64-71;  Kimball,  §§389-391;  Reed  &  Guthe,  §159; 
Spinney,  §§165-166;  Watson,  §§190-192. — More  Advanced:  Poynting  & 
Thomson  (Heat),  Chap.  Ill;  Watson  (Pr.},  §§73-77. 

The  object  of  this  experiment  is  to  determine  the  coefficient 
of  apparent  expansion  of  some  salt  solution  with  reference  to 
glass.  A  vessel  holds  M  grams  of  liquid  at  t°  and  m  grams  at  a 
higher  temperature,  t'°.  Let  V  be  the  volume  of  the  vessel  at 
the  lower  temperature.  Since  we  are  considering  the  apparent 
expansion,  i.  e.,  the  expansion  with  reference  to  the  vessel,  we 
may  consider  V  to  be  also  the  volume  of  the  vessel  at  the  higher 
temperature.  The  volume  of  I  gram  at  t°  is  therefore  V/M 
and  at  t'°,  V/m.  The  increase  in  volume  is 

V     V 


m    M        \  Mm 

The  coefficient  of  apparent  expansion,  e,  is  this  apparent  ex- 
pansion divided  by  the  original  volume  V/M  and  the  range  of 
temperature  (/' '  — /)  or 

M—m 

~m(t'-ty 

A  glass  bulb  with  a  re-curved  capillary  stem  is  used.  To  fill 
the  bulb  with  a  liquid,  warm  it  with  the  hand  or  by  playing  a 
flame  about  some  distance  beneath  it.  Remove  it  from  the 
source  of  heat  and  plunge  the  end  of  the  stem  into  the  liquid. 
As  the  air  in  the  bulb  cools  liquid  will  be  drawn  in. 

To  expel  liquid,  warm  the  bulb  gently,  keeping  it  so  turned 
that  the  stem  is  filled  with  the  liquid;  when  the  liquid  ceases 
to  come  out,  invert  it  so  that  the  stem  is  highest,  and  allow  it 
to  partially  cool.  Repeat  until  all  the  liquid  is  expelled. 

Clean  the  bulb  by  drawing  in  a  little  distilled  water,  or,  if  the 
interior  be  foul,  first  use  chromic  acid.  Finally  rinse  the  interior 
with  alcohol.  Remove  the  alcohol  and  dry  the  interior,  if  neces- 
sary playing  a  flame  about  some  distance  beneath. 


70  HEAT. 

To  determine  the  density  of  the  (cold)  salt  solution,  thoroughly 
cleanse  a  tall  measuring  glass  and  a  suitable  hydrometer  (variable 
immersion).  Pour  enough  of  the  salt  solution  into  the  measur- 
ing glass  to  float  the  hydrometer,  read  the  density,  and  pour  the 
solution  back  into  the  bottle. 

Weigh  the  bulb  very  carefully  on  a  sensitive  balance  (see  pp. 
20-23).  Support  the  bulb  in  a  clamp  stand,  clasping  the  stem 
between  half  corks.  Fill  a  small  beaker  with  the  salt  solution 
and  support  it  so  that  the  end  of  the  stem  dips  into  the  solution. 
Warm  the  bulb,  playing  a  Bunsen  flame  beneath.  Never  allow 
the  flame  for  an  instant  to  remain  stationary  beneath  the  bulb,  and 
until  the  bulb  contains  considerable  warm  liquid,  do  not  allow  the 
flame  to  touch  the  bulb,  and  then  only  where  there  is  liquid.  Alter- 
nately warm  the  bulb  and  allow  it  to  cool  a  little  until  the  bulb  is 
filled.  When  it  is  partly  full  it  may  be  best  to  gently  boil  the 
liquid  in  the  bulb.  When  the  bulb  is  almost  full  the  liquid  can  be 
made  to  expand  to  fill  the  entire  stem.  Then  allow  it  to  cool 
completely  while  it  draws  over  liquid  from  the  beaker. 

When  the  bulb  is  cooled  to  the  temperature  of  the  room,  sup- 
port it  in  a  copper  vessel  in  which  water  is  kept  at  a  constant 
temperature,  a  few  degrees  warmer  than  the  room.  When  the 
temperature  has  been  kept  constant  for  five  minutes  (by  the 
addition  of  small  amounts  of  hot  or  cold  water,  if  necessary) 
and  has  been  frequently  stirred,  read  the  temperature  (as  always 
estimating  tenths).  Remove  any  liquid  adhering  to  the  end  of 
the  stem,  remove  the  bulb  from  the  bath,  dry  the  exterior,  and 
weigh.  Handle  the  bulb  carefully  with  a  cloth  about  it  so  that  no 
liquid  may  be  expelled.  Weigh  a  small,  clean,  dry  beaker. 
Support  the  bulb  again  in  the  copper  bath  with  the  beaker  be- 
neath the  end  of  the  stem,  to  catch  any  liquid  expelled.  Heat 
the  water  in  the  bath  to  boiling.  When  the  temperature  has 
been  constant  for  five  minutes,  read  the  temperature,  catch 
on  the  side  of  the  small  beaker  any  liquid  adhering  to  the  end  of 
the  stem,  remove  the  bulb  from  the  bath,  dry  the  exterior,  and 
weigh.  Weigh  the  small  beaker  with  the  liquid  contained. 
Carefully  remove  the  liquid  from  the  bulb  and  stem  as  described 
above. 

The  difference  between  the  two  weights  of  the  bulb  when 


COEFFICIENT  OF  INCREASE  OF  PRESSURE  OF  AIR.      7 1 

filled  with  liquid  gives  the  weight  M—m  of  liquid  expelled. 
The  difference  between  the  weight  of  the  flask  dry  and  after  being 
in  the  second  bath  gives  the  final  weight  of  liquid  in  the  bulb. 
The  expelled  liquid  is  saved  simply  as  a  check  and  is  not  used 
at  all  if  the  above  difference  be  slightly  greater. 

A  specific-gravity  bottle  may  be  substituted  for  the  bulb,  but 
is  not  as  satisfactory. 

Questions. 

1.  Why  is  double  weighing  unnecessary? 

2.  Why  is  M — m  determined  more  accurately  from  the  difference  of  the 
two  weighings  than  from  the  weight  of  the  liquid  expelled? 

3.  How  might  the  coefficient  of  expansion  of  a  solid,  attainable  only  in  the 
form  of  small  lumps,  be  found  by  an  extension  of  this  method? 

4.  How  might  the  absolute  expansion  of  a  liquid  be  found  by  the  above 
apparatus? 


XVI.  COEFFICIENT  OF  INCREASE  OF  PRESSURE  OF 

AIR. 

References — Elementary:  Duff,  §§265,  279;  Ames,  p.  240;  Crew,  §269;  Edser, 
(Heat),  pp.  106-111 ;  Kimball,  §§393-397;  Reed  &  Guthe,  §163;  Spinney, 
§§157,  161;  Watson,  §§195-198. — More  Advanced:  Poynting  Of  Thomson 
(Heat),  Chap.  IV;  Watson  (Pr.),  §§78-79. 

If  the  volume  of  a  mass  of  gas  remains  constant  while  its 
temperature  is  raised,  its  pressure  increases  according  to  the  law 

P  *=  Po  (i  +  a  0, 

in  which  P0  is  the  pressure  at  o°  C.,  P  the  pressure  at  the  tempera- 
ture /,  and  a  is  a  constant  called  the  coefficient  of  increase  of  pressure. 
If  the  pressure  were  kept  constant  and  the  volume  allowed  to 
increase,  the  law  of  increase  of  volume  would  be  similar,  and  it  is 
found  that  the  constant  a  is  practically  the  same  in  both  cases. 

It  is,  however,  difficult  to  keep  the  volume  exactly  constant, 
for  the  containing  vessel  will  expand  when  heated  (the  volume 
of  the  vessel  would  also  increase  because  of  the  increase  of  pres- 
sure to  which  it  is  subjected,  but  this  may  be  neglected  since  it  is 
extremely  small).  We  can,  however,  allow  for  the  expansion  of 
the  vessel.  For,  by  the  general  gas  law, 

P  v  =  P0Fo  (i  +  at) 


72  HEAT. 

and  the  law  of  expansion  of  the  vessel  is 


where  b  is  the  coefficient  of  cubical  expansion  of  the  vessel 
Dividing  the  first  equation  by  the  second  and  neglecting  the  pro 
ducts  of  small  quantities,  we  get 


Hence, 


The  air  (or  gas)  is  enclosed  in  a  bulb  to  which  is  connected  a 
mercury  manometer.  The  pressure  indicated  by  the  mano- 
meter is  obtained  from  readings  of,  the  mercury  levels  on  a  scale 
between  the  two  columns,  or,  preferably,  with  a  cathetometer 

(P-  17). 

If  the  true  increase  of  pressure  of  dry  air  is  desired  the  air  must 
first  be  carefully  dried.     To  fill  the  bulb  with  dry  air  it  may  be 
connected  through  a  drying  tube  (containing 
chloride  of  calcium)  with  an  air-pump  and  the 
bulb  several  times  exhausted  and  refilled  with 
f,          -=\     |     air  sucked  through  the  drying  tube.     (If  the 
()  bulb  be  already  filled  with  dry  air  the  process 

\       i  will  be  unnecessary.) 

fc  The  bulb  is  then  connected  to  the  manometer. 

The  bulb  is  first  immersed  in  a  bath  of  ice  and 
water  as  nearly  as  possible  at  o°,  and  the  mov- 
able column  of  the  manometer  is  adjusted  until 
the  mercury  in  the  other  column  is  at  a  definite 
point,  as  high  as  possible  without  entering  the  contraction  where 
connection  is  made  with  the  capillary.  The  temperature  and 
pressure  are  read  as  carefully  as  possible,  at  least  six  times,  when 
both  have  become  quite  steady,  the  manometer  being  readjusted 
before  each  reading. 

The  bath  of  ice  and  water  is  now  replaced  by  one  of  water  at 
about  10°  and  the  movable  column  is  readjusted  until  the  mercury 
in  the  stationary  column  is  at  the  former  point,  that  the  volume 


FIG.  21. 


COEFFICIENT  OF  INCREASE  OF  PRESSURE  OF  AIR.      73 

of  the  gas  may  remain  constant.  The  temperature  and  pressure 
are  read  when  they  have  become  steady.  The  water  is  then 
heated  to  about  20°  and  the  observations  are  repeated.  Read- 
ings are  thus  made  at  intervals  of  about  10°  until  the  water  boils. 

The  pressure  and  temperature  when  the  water  is  boiling  should 
be  read  at  least  six  times,  the  mercury  level  in  the  stationary 
column  being  adjusted  to  the  constant  point  before  each  reading. 
It  is  at  the  initial  temperature  (near  o°)  and  the  final  temperature 
(near  100°)  that  the  most  reliable  observations  are  obtained, 
and  it  is  upon  these  that  the  most  reliable  estimate  of  the  coeffi- 
cient of  expansion  is  founded.  (The  readings  at  intermediate 
temperatures  are  made  in  order  to  test  the  law  of  expansion.) 
If  the  two  arms  of  the  manometer  are  of  different  radii,  there 
will  be  a  constant  difference  of  level  due  to  capillarity.  This 
should  be  read  when  the  bulb  is  disconnected  and  allowance 
should  be  made  for  it  at  other  times.  Read  the  barometer  (p.  19) 
and  the  temperature  of  the  barometer  and  of  the  mercury  in  the 
manometer. 

Tabulate  from  your  observations  (a)  the  temperatures;  (b) 
the  differences  in  level  of  the  mercury  columns;  (c)  these  differ- 
ences reduced  to  zero  degrees ;  (d)  the  pressures  as  calculated  from 
(c)  and  the  barometer  heights  (reduced  to  zero  degrees). 

The  test  of  the  law  of  expansion  is  made  by  plotting  the  curve 
of  pressure  and  temperature,  the  former  as  ordinates,  the  latter 
as  abscissas.  This  should  be  nearly  a  straight  line.  The  aver- 
ages for  the  first  point  (o°)  and  the  last  (about  100°)  are  to  be 
taken  as  fixing  the  straight  line.  The  divergence  of  intermediate 
points  from  the  straight  line,  while  not  sufficient  to  invalidate 
the  conclusion  that  the  increase  of  pressure  is  linear,  will  illustrate 
the  difficulty  of  keeping  the  temperature  at  intermediate  points 
constant  for  a  sufficient  length  of  time  for  the  air  in  the  bulb  to 
come  wholly  to  the  temperature  of  the  water. 

Calculate  from  these  two  average  pressures  and  temperatures, 
the  coefficient  of  apparent  increase  of  pressure  (af  =  (P  —  Po)/Pot), 
and,  obtaining  the  coefficient  of  cubical  expansion  of  the  glass  (b) 
from  Table  VIII,  find  the  true  coefficient  of  increase  of  pressure 
(a).  (Remember  that  the  coefficient  of  cubical  expansion  is  three 
times  the  coefficient  of  linear  expansion.) 


74 


HEAT. 


If  time  permit,  increase  the  range  of  temperature  by  observa- 
tions below  o°  in  a  freezing  mixture  and  above  100°  in  heated  oil. 
Also,  use  the  apparatus  as  an  air  thermometer  to  determine  the 
room  temperature. 

Questions. 

1.  Why  must  (a)  the  air  be  dry?     (b)  the  bulb  be  dry?     (c)  a  capillary 
connect  the  bulb  and  the  manometer? 

2.  What  would  be  the  percentage  error  if  the  expansion  of  the  bulb  was 
neglected? 

3.  What  sources  of  error  remain  uncorrected? 


XVII.   PRESSURE  OF  SATURATED  WATER  VAPOR. 

References — Elementary:  Duff,  §§308,  310-3121  Ames,  pp.  264-269;  Crew,  §§279- 
281;  Edser  (Heat),  pp.  220-228;  Kimball,  §§433~435;  Reed  &  Guthe,  §§199- 
202;  Spinney,  §187;  Watson,  §§216-218. — More  Advanced:  Poynting  & 
Thomson  (Heat),  pp.  172-175. 

The  object  of  this  experiment  is  to  find  the  pressure  of  saturated 
water  vapor  at  different  temperatures.  By  pressure  of  saturated 
water  vapor  at  a  given  temperature,  or,  as  it  is  often  called, 
maximum  pressure  of  water  vapor,  or,  equilibrium 
pressure,  is  denoted  the  pressure  of  the  vapor  above 
water  in  a  closed  vessel  at  the  given  temperature 
after  a  steady  state  has  been  reached.  A  liquid 
continues  to  give  off  vapor  from  the  surface,  or, 
"evaporate"  as  long  as  the  pressure  of  the  vapor 
above  the  liquid  is  less  than  the  saturated  vapor 
pressure,  independent  of  the  total  atmospheric 
pressure  above  the  liquid.  After  the  pressure  of 
the  vapor  reaches  the  saturated  vapor  pressure  for 
that  temperature,  the  total  quantity  of  vapor  in 
the  atmosphere  above  the  liquid  remains  constant, 
since  for  any  vapor  given  off  from  the  surface  an 
equal  quantity  is  condensed. 

There  are  two  chief  methods  of  finding  the  satur- 
ated vapor  pressure,  the  static  method  and   the 
kinetic  method. 
(A)  In  the  static  method  some  water  (or  other  liquid)  is  intro- 
duced into  the  space  at  the  top  of  a  barometric  column  which  is 
surrounded  by  a  bath,  the  temperature  of  which  can  be  varied. 


FIG.  22. 


PRESSURE    OF   SATURATED   WATER   VAPOR. 


75 


The  pressure  of  the  vapor  is  found  by  measuring  with  a  catheto- 
meter  (p.  17)  the  height  of  the  mercury  column  and  subtracting 
this  from  the  barometric  reading,  each  being  reduced  to  zero 
(p.  19).  By  varying  the  temperature  of  the  bath,  the  vapor 
pressure  at  various  temperatures  is  obtained. 

(B)  In  the  kinetic  method  the  quantity  measured  is  the  tempera- 
ture of  the  steam  above  water  boiling  under  different  measured 
pressures.  When  a  liquid  boils,  bubbles  of  vapor  are  formed 
throughout  the  interior  of  the  liquid.  In  forming  these  bubbles, 
the  vapor  overcomes  the  pressure  of  the  atmosphere  above  the 
liquid,  therefore  the  pressure  of  the  vapor  must  equal  the  at- 
mospheric pressure,  and  obviously  the  vapor  in  the  bubbles  is 


FIG.  23. 

saturated.  Hence,  in  measuring  the  atmospheric  pressure  above 
a  liquid  boiling  at  a  known  temperature,  we  find  the  saturated 
vapor  pressure  of  the  liquid  at  this  temperature,  and  this  is 
Regnault's  method,  which  method  is  followed  in  this  experiment. 
In  Regnault's  apparatus  the  total  pressure  above  the  surface 
of  the  liquid  can  be  kept  very  constant.  As  the  liquid  is  heated, 
the  vapor  is  condensed  in  a  Liebig  condenser,  and  as  the  pressure 
of  vapor  distributed  through  several  conducting  vessels  is  the 
vapor  pressure  corresponding  to  the  vessel  at  lowest  tempera- 
ture, the  pressure  exerted  by  the  vapor  cannot. exceed  the  maxi- 
mum pressure  corresponding  to  the  temperature  of  the  tap  water, 
and  is  therefore  very  small.  As  the  temperature  of  the  boiler 


76  HEAT. 

changes,  the  temperature  of  the  air  in  the  boiler  varies,  but  a 
large  air  reservoir  surrounded  by  water  is  connected  between  the 
condenser  and  the  manometer  and  air-pump  or  aspirator,  which 
makes  the  volume  of  the  air  in  the  boiler  small  compared  with  the 
total  volume  of  air  in  the  system,  and  thus  the  increase  of  pres- 
sure due  to  the  heating  of  the  air  in  the  boiler  is  small. 

The  boiler  should  be  about  two-thirds  full  of  water.  Fill  with 
water  the  small  tube  running  down  into  the  boiler  (which  tube  is 
closed  at  the  bottom) ,  and  insert  in  this  tube  through  a  cork  one 
of  the  thermometers  tested  by  the  observer.  Draw  out  any 
water  which  may  be  in  the  air  reservoir.  Fill  the  surrounding 
vessel  with  water.  (Rubber  stoppers  should  be  lubricated  with 
rubber  grease  (note  p.  29))  before  insertion.  Exhaust  the  air 
from  the  system  to  the  highest  vacuum  attainable  by  means 
of  a  Geryk  pump  or  aspirator.  Close  all  the  cocks  through  which 
connection  is  made  to  the  aspirator  and  let  the  system  stand  a 
few  minutes  to  see  if  there  is  any  leakage.  If  not,  start  a  gentle 
stream  of  water  through  the  condenser,  and  place  a  Bunsen  flame 
under  the  boiler.  Read  the  barometer  and  its  temperature  (see 
p.  19). 

When  the  temperature  as  registered  by  the  thermometer  in 
the  boiler  becomes  very  steady,  read  it,  and  at  once  record  the 
two  extremities  of  the  mercury  column  of  the  manometer.  Let 
in  a  little  air.  If  there  are  two  cocks,  first  open  and  then  close 
the  cock  near  the  air-pump  and  then  open  and  close  the  cock  nearer 
the  apparatus.  Increase  the  pressure  at  first  by  about  15  mm., 
gradually  increasing  the  steps  and  when  near  atmospheric  pres- 
sure change  the  pressure  by  about  12  cm.  The  reason  for  the 
difference  in  pressure  in  the  steps  is  that  it  is  better  to  have  the 
steps  represent  about  equal  changes  of  temperature,  for  instance, 
about  5°. 

From  the  corrected  barometer  reading  and  the  differences  in 
height  of  the  mercury  columns,  calculate  the  pressures.  Tabu- 
late pressures  and  temperatures  and  also  plot  them,  making 
temperatures  abscissas  and  pressures  ordinates. 

Ramsay  and  Young's  method  for  measuring  the  vapor  pressure 
of  a  small  quantity  of  liquid  is  described  in  Watson  s  Practical 
Physics,  §94. 


HYGROMETRY.  77 

Questions. 

1.  State  precisely  what  two  quantities  you  have  observed  in  the  second 
method  and  what  relation  they  bear  to  the  pressure  and  temperature  of  satur- 
ated vapor. 

2.  What  condition  determines  whether  a  liquid  will  boil  or  evaporate  at 
a  given  temperature? 

3.  What  was  the  actual  vapor  pressure  (a)  in  the  bubbles  rising  to  the  sur- 
face (&)  in  the  space  above  the  boiling  liquid?     (Table  XIII.) 

4.  What  determines  (a)  the  lowest  temperature,  (&)  the  highest  temperature 
for  which  this  apparatus  is  applicable? 


XVIII.  HYGROMETRY. 

References — Elementary:  Duff,  §309;  Ames,  pp.  265-268;  Kimball,  §§447-448; 
Reed  &  Guthe,  §§213-215;  Spinney,  §§193-197;  Watson,  §§220-221. — 
More  Advanced:  Davis'  Elementary  Meteorology,  Chap.  VIII;  Poynting  & 
Thomson  (Heat),  pp.  209-215;  Watson  (Pr.},  §§95-97. 

Three  methods  will  be  used  for  studying  the  hygrometric  state 
of  the  atmosphere.  The  first  method  (A)  determines  the  dew- 
point,  the  second  (B)  determines,  indirectly,  the  actual  vapor 
pressure,  and  the  third  (C)  determines  the  relative  humidity. 

(A)  Regnault's  Hygrometer. — A  thin  silvered-glass  test-tube 
is  half-filled  with  ether.  The  test-tube  is  tightly  closed  by  a  cork 
through  which  passes  a  sensitive  thermometer  which  gives  the 
temperature  of  the  ether.  Two  glass  tubes  also  pass  through  the 
cork,  one  extending  to  the  bottom,  the  other  ending  below  the 
cork.  An  aspirator  gently  draws  air  from  the  shorter  tube. 
The  ether  is  evaporated  by  the  air  bubbles  and  the  entire  vessel 
cools.  The  silvered  surface  and  the  thermometer  are  watched 
through  a  telescope  and  the  temperature  is  read  the  moment 
moisture  appears  on  the  metal.  The  air  current  is  stopped  and 
the  temperature  of  disappearance  of  the  moisture  is  observed. 
This  is  repeated  several  times  and  the  mean  is  taken  as  the  dew- 
point.  The  detection  of  moisture  is  facilitated  by  observing 
at  the  same  time  a  similar  piece  of  silvered  glass  which  covers 
a  part  of  the  test-tube,  but  which  is  insulated  from  it.  The 
temperature  of  the  air  should  also  be  carefully  determined, 
preferably  with  a  thermometer  in  a  similar  apparatus  where  there 
is  no  evaporation. 

An  arrangement  of  two  small  mirrors  at  right  angles  so  placed 


78  HEAT. 

as  to  reflect  light  from  the  two  tubes  into  the  telescope  will 
facilitate  the  comparison. 

(B)  Wet  and  Dry  Bulb  Hygrometer. — Two  thermometers  are 
mounted  a  few  inches  apart.     About  the  bulb  of  one  is  wrapped 
muslin  cloth  to  which  is  attached  a  muslin  wick  dipping  in  water. 
The  other  is  bare.     The  temperatures  of  both  are  read  when  they 
have  become  steady.     The  temperature  of  the  first  thermometer 
will  be  lower  than  that  of  the  bare  thermometer,  on  account  of 
the  evaporation  of  the  water.     From  the  difference  of  tempera- 
ture of  the  two  thermometers  and  the  temperature  of  the  bare 
thermometer  the  actual  vapor  pressure  may  be  determined  with 
the  aid  of  empirical  tables  (see  Table  XV).     For  more  accurate 
apparatus,  see  references. 

(C)  Chemical  Hygrometer. — Fill  three  ordinary  balance  drying 
vessels  with  pumice.     Saturate  two  with  strong  sulphuric  acid 
and  the  third  with  distilled  water.     Weigh  very  carefully  the 
two  which  have  the  acid  and  then  connect  them  to  an  aspirator, 
with  the  water  absorption  vessel  between  them.     After  a  gentle 
stream  of  air  has  passed  through  for  a  considerable  time,  dis- 
connect and  weigh  the  sulphuric  acid  vessels.     The  ratio  of  the 
gains  in  weight  will  obviously  be  the  relative  humidity.     Observe 
also  the  temperature  of  the  air. 

If  not  directly  determined,  calculate  from  your  observation, 
by  each  of  the  three  methods,  the  dew-point,  the  actual  vapor 
pressure,  the  relative  humidity,  and  the  amount  of  moisture  in 
the  atmosphere  per  cubic  meter.  Tabulate  your  results.  Table 
XIII  gives  the  vapor  pressures  of  water  at  different  temperatures. 


XIX.    SPECIFIC  HEAT  BY  THE  METHOD  OF  MIXTURE. 

References — Elementary:  Duff,  §§281-284;  Ames\  pp.  250-252;  Crew,  §252; 
Edser  (Heat),  pp.  122-136;  Kimball,  §401;  Reed  &  Guthe,  §§168-173; 
Spinney,  §§168-172;  Watson,  §§200-201. — More  Advanced:  Poynting  & 
Thomson  (Heat),  pp.  66-69. 

The  specific  heat  of  a  substance  is  the  number  of  calories  re- 
quired to  raise  the  temperature  of  one  gram  of  the  substance  one 
degree  centigrade,  or  the  number  of  calories  given  up  by  one 
gram  in  cooling  one  degree  centigrade.  In  the  method  of  mix- 


SPECIFIC   HEAT    BY   THE   METHOD   OF   MIXTURE.  79 

ture  a  known  mass  (M)  of  the  substance,  heated  to  a  known 
temperature  (T),  is  immersed  in  a  known  mass  of  liquid  (m) 
of  known  specific  heat  (for  water  =  i),  at  a  known  temperature 
(to),  and  the  unknown  mean  specific  heat  (x)  of  the  substance  is 
deduced  from  these  data  and  the  temperature  (/)  to  which  the 
mixture  rises.  Water  is  the  liquid  employed  unless  there  would 
be  a  chemical  reaction  on  immersion. 

The  liquid  must  be  contained  in  a  vessel  which  is  also  heated 
by  the  immersion  of  the  hot  body.  The  heating  of  the  vessel 
is  equivalent  to  the  heating  of  a  certain  additional  quantity  of 
water.  This  equivalent  quantity  of  water  (e)  is  called  the  water 
equivalent  of  the  vessel.  It  is  practically  equal  to  the  mass  of 
the  vessel  (mi)  multiplied  by  the  specific  heat  (s)  of  the  material 
of  the  vessel.  Theoretically  it  may  be  obtained  by  noting  the 
temperature  of  the  vessel  and  pouring  into  it  a  known  mass  of 
water  at  a  known  temperature  and  noting  the  final  temperature. 
This  is  an  inverted  form  of  the  method  of  mixture  applied  to 
finding  the  specific  heat  of  the  vessel.  But  as  we  shall  presently 
see,  it  is  the  method  of  mixture  applied  under  very  unfavorable 
conditions  and  will  not  usually  give  a  very  satisfactory  result. 
Another  method  will  be  recommended  below. 

The  equation  for  finding  the  specific  heat  is  obtained  by  equat- 
ing the  heat  given  up  by  the  hot  body  to  that  taken  up  by  the 
water  and  containing  vessel.  Hence 

Mx(T-t)  =  (m+e)(t-t0). 

Sources  of  Error.  * 

(1)  Losfe  of  heat  while  the  hot  body  is  being  transferred  to  the 
water. 

(2)  Loss   of  heat   by   radiation,   conduction,   or  evaporation 
while  the  mixture  is  assuming  a  uniform  temperature. 

(3)  Errors    in   ascertaining   the    true    temperature    including 
errors  in  the  thermometers. 

Choice  of  Best  Conditions. — As  the  accuracy  of  this  determina- 
tion depends  largely  on  the  selection  of  suitable  condition^,  we 
shall  consider  how  these  may  be  chosen  so  that  unavoidable 
errors  in  the  separate  measurements  may  affect  the  result  as 
little  as  possible. 


8o  HEAT. 

By  taking  the  logarithms  of  both  sides  of  (i)  and  differentiat- 
ing partially,  we  obtain  (see  p.  7.) 

(2)  (3)  (4) 


x_M~      M'      _xjm~m+e'     _x    e~  m+e' 
(5)  (6)  (7) 

dT    i~5*~|       g/0    r&ri     (T-^t 

X_T      T-?'  [_x_\to      /-A,1   L*J*   (T-t)(t-toY 

The  left-hand  side  of  (2)  stands  for  "the  ratio  that  the  possible 
error  (5x)  in  x,  due  to  the  possible  error  (8M)  in  M,  bears  to  x" 
and  so  for  the  other  equations. 

M  and  m  can  be  measured  with  great  precision;  hence  (2) 
and  (3)  are  negligible.  From  (4)  it  is  seen  that  the  water  equiva- 
lent of  the  calorimeter  must  be  found  with  some  care.  From 
(5)  and  (6)  it  is  seen  that  the  ranges  T—t  and  t  —  to  should  be  as 
great  as  possible  (see,  however,  "sources  of  error"  above). 
This  is  also  consistent  with  the  indications  of  (7),  for  although 
(T  —  to)  enters  the  numerator,  the  product  of  T—t  and  t  —  to 
is  in  the  denominator.  Moreover,  it  is  seen  from  (5),  (6),  and 
(7)  that  if  equal  errors  are  made  in  observing  T,  t,  and  to,  the 
effect  of  the  error  in  t  may  equal  the  sum  of  the  effects  of  the 
errors  in  T  and  to.  Hence  the  necessity  of  determining  t  with 
special  care.  But,  allowing  an  unavoidable  error  in  /,  how  can 
its  effect  be  made  as  small  as  possible  by  properly  choosing  the 
quantities,  M,  m,  to,  T?  Let  us  suppose  T—  to  is  taken  as  great 
as  possible  under  the  circumstances.  How  can  (T  —  t)  X  (t  —  to) 
be  made  as  great  as  possible?  The  sum  of  T  —  t  and  t  —  to  is 
T  —  to,  a  fixed  quantity.  Hence  their  product  is,  by  algebra,  a 
maximum  when  they  are  equal  or  /  is  midway  between  T  and  to. 
But  it  is  seen  from  (i)  that  this  also  requires  MX  and  m  +  e  to 
be  equal.  Hence  we  see  that  for  the  best  results,  T  and  t0  should 
be  as  far  apart  as  possible  and  the  heat  capacity  of  the  specimens 
should  be  as  nearly  as  possible  equal  to  the  heat  capacity  of  the  water 
and  the  vessel  that  contains  it. 

The  logical  procedure,  then,  would  be  to  roughly  determine  x 
by  the  method  of  mixture,  using  any  convenient  values  of  M  and 


SPECIFIC   HEAT    BY   THE    METHOD    OF   MIXTURE.  8 1 

m,  and  with  this  rough  value  for  x,  calculate  what  ratio  of  M 
to  m  would  best  satisfy  the  above  condition.  Moreover,  it  is 
seen  from  (4)  above  that  m  should  be  as  large  as  is  consistent 
with  other  conditions.  Then  we  should  proceed  to  arrange  a 
new  experiment  to  be  performed  under  the  more  favorable  con- 
ditions for  precision. 

We  now  see  why  it  is  not  easy  to  determine  the  water  equivalent 
of  the  vessel  directly.  Its  heat  capacity  is  small  compared  with 
that  of  the  water  that  would  fill  it,  and  so  the  change  of  the 
temperature  of  the  water  would  be  small  and  difficult  to  deter- 
mine accurately.  If  a  much  smaller  quantity  of  water  were  used, 
a  large  part  of  the  surface  of  the  vessel  would  be  left  uncovered, 
and  its  temperature  could  not  be  determined.  Hence  it  is  better 
to  determine  the  specific  heat  of  the  material  of  the  vessel,  using 
the  ordinary  method  of  mixture  and  a  mass  of  the  same  material 
as  the  vessel.  Then  multiplying  the  mass  of  the  vessel  by  its 
specific  heat,  we  have  its  water  equivalent. 

It  is  desirable  that  the  body  should  have  such  a  form  that  it 
and  the  water  in  which  it  is  immersed  should  rapidly  come  to  a 
common  temperature.  Filings,  shot,  thin  strips,  wire  or  small 
pieces  would  best  satisfy  this  condition.  Larger  solid  masses 
are  more  rapidly  (and  therefore  with  less  loss  of  temperature) 
transferred  from  the  heater  to  the  water,  and  to  give  the  water 
ready  access  to  them  they  may  be  perforated  with  holes,  through 
which,  by  moving  the  mass  up  and  down  in  the  water,  the  water 
may  be  made  to  circulate.  The  following  directions  apply 
primarily  to  this  latter  form,  but  may  be  readily  adapted  to  the 
other  forms. 

Two  forms  of  heater  will  be  here  described,  (i)  The  steam 
heater.  A  copper  tube  large  enough  to  admit  the  specimen  is 
enclosed,  except  at  the  ends,  by  an  outer  copper  vessel  which  is 
to  act  as  a  steam  jacket  to  the  inner  vessel.  Steam  from  a  simple 
form  of  boiler  is  admitted  to  the  jacket  through  a  tube  near  the 
top  of  the  jacket  and  escapes  through  an  outlet  near  the  bottom. 
If  the  body  to  be  heated  is  a  solid  mass,  it  is  suspended  in  the 
heater  by  a  long  string  that  passes  through  a  cork  that  closes 
the  upper  end  of  the  heater.  (If  the  specimen  is  in  the  form  of 
shot  or  clippings  they  are  placed  in  a  dipper  that  fits  into  the 
6 


82  HEAT. 

heater.)  A  thermometer  passed  through  the  cork  or  cover 
of  the  dipper  is  pushed  down  until  it  comes  into  contact  with  the 
body  tested.  The  lower  end  of  the  heater  is  also  closed  with  a  cork. 

(2)  The  Electric  Heater. — A  metallic  tube  is  heated  by  a  strong 
current  of  electricity  passing  through  a  coil  of  wire  of  high  re- 
sistance that  surrounds  the  tube.  The  current  can  be  varied 
by  changing  a  variable  resistance  in  circuit  with  the  heating  coil. 
With  an  alternating  current  the  resistance  may  be  an  inductive 
resistance  or  choking  coil  consisting  of  wire  surrounding  a  soft- 
iron  wire  core.  A  low  resistance  allowing  a  high  current  is  used 
until  the  temperature  rises  to  the  desired  point  (perhaps  near 
1 00°)  and  then  the  current  is  reduced  to  the  strength  that  will 
keep  the  temperature  constant,  as  indicated  by  a  thermometer 
hung  in  the  heater.  The  body  is  introduced  into  the  heater 
exactly  as  in  the  case  of  the  steam  heater.  The  proper  method 
of  varying  the  resistance  can  only  be  learned  by  some  practice. 

It  is  convenient  to  have  a  slot  in  the  cover  of  either  heater  and 
above  the  slot  a  multiple  spring  clip  which  will  hold  a  ther- 
mometer vertical  at  any  point  in  the  slot.  As  an  aid  in  securing 
a  uniform  temperature,  one  thermometer  may  be  hung  within 
the  specimen  and  the  other  just  outside.  The  specimen  should 
be  hung  as  low  as  possible  in  the  heater  that  the  stem  correction 
(p.  65)  of  the  thermometer  may  be  negligible. 

The  calorimeter  may  be  prepared  while  the  specimen  is  being 
heated.  It  consists  of  a  smaller  copper  or  aluminum  can  highly 
polished  on  the  outside  and  enclosed  in  a  larger  one  brightly 
polished  on  the  inside,  but  well  insulated  from  it  by  corks  or 
cotton-wool.  A  wooden  cover  fits  over  both  vessels  and  has 
holes  for  thermometer  and  stirrer  and  an  opening  giving  access 
to  the  interior  vessel.  A  convenient  form  has  a  trap-door  which 
slides  open  in  two  halves,  exposing  the  entire  inner  vessel.  A 
screen  with  sliding  or,  preferably,  double  swinging  doors,  should 
separate  the  calorimeter  from  the  boiler,  heater,  etc. 

The  "water  equivalent"  of  the  receiving  vessel  means  the  water 
equivalent  of  the  inner  vessel  together  with  the  stirrer,  if  one  be 
used.  It  is  advisable  that  the  stirrer  should  also  be  of  copper 
or  aluminum.  (A  stirrer  is,  however,  not  necessary  when  the 
specimen  is  in  the  form  of  one  large  block  (see  p.  81).) 


SPECIFIC   HEAT    BY   THE   METHOD   OF   MIXTURE.  83 

At  certain  times,  in  the  manipulation  of  this  experiment  the 
co-operation  of  two  persons  is  desirable,  and,  for  economy  of 
time,  two  determinations  should  be  made  simultaneously,  two 
heaters,  two  specimens,  and  two  calorimeters  being  used.  One 
specimen  should  preferably  be  of  the  same  material  as  the  calori- 
meter, so  that  the  water  equivalent  may  be  determined. 

The  body  whose  specific  heat  is  to  be  determined  is  weighed 
to  o.i  gm.  and  placed  in  the  heater  along  with  a  thermometer. 
The  inner  vessel  of  the  calorimeter  (including  the  stirrer)  is 
weighed  to  o.i  gm.  Water  near  the  temperature  of  the  room  is 
poured  into  it  until  it  is  judged  that  when  the  hot  body  is  im- 
mersed it  will  be  completely  covered  and  the  water  will  rise  to 
within  a  couple  of  centimeters  of  the  top  of  the  vessel.  The 
vessel  and  water  are  then  weighed  to  o.i  gm.  The  inner  vessel 
is  now  replaced  in  the  outer  and  the  cover  adjusted  and  closed. 

When  the  temperature  of  the  specimen  has  remained  constant 
for  ten  minutes,  it  may  be  assumed  that  the  hot  body  is  through- 
out at  the  temperature  of  the  heater.  The  next  steps  require 
two  persons,  and  as  it  is  important  that  it  should  be  carried  out 
promptly  and  neatly,  it  should  be  carefully  considered  before 
being  performed.  One  person  constantly  stirs  the  water  in  the 
calorimeter  and  reads  the  temperature,  to  tenths  of  a  degree, 
every  minute  for  five  minutes.  He  then  opens  the  cover  and 
slides  the  calorimeter  beneath  the  heater.  The  other  observer 
has  meanwhile  made  a  careful  final  observation  of  the  tempera- 
ture of  the  heater  and  removed  the  lower  stopper.  As  soon  as 
the  calorimeter  is  in  position,  he  lowers  the  hot  body,  without 
splash,  into  the  water.  The  calorimeter  must  then  be  immedi- 
ately removed  and  the  cover  closed. 

One  observer  should  then  keep  the  mixture  stirred  by  moving 
the  body  up  and  down  with  the  aid  of  the  string  and  note  the 
temperature  at  as  short  equal  intervals  as  possible  (perhaps  every 
15  sees.)  while  the  other  records  the  readings.  After  the  highest 
temperature  has  been  reached,  the  readings  are  continued  every 
minute  for  at  least  five  minutes. 

Simple  and  obvious  modifications  of  the  above  procedure  are 
required  if  the  specimen  is  in  the  form  of  shot  or  clippings. 

After  rough  calculations  of  water  equivalents  and  specific 
heats,  the  observers  should  exchange  duties  and  repeat  the 


84  HEAT. 

whole,  using  masses  that  most  nearly  accord  with  the  conditions 
laid  down  in  the  considerations  stated  above.  Plot  all  the 
temperature  readings  and  from  the  smooth  portions  of  the  curve, 
determine  t  and  to,  correcting  for  radiation  if  the  time  of  rise  of 
temperature  was  appreciable  (see  pp.  59-61). 

The  possible  error  of  the  result  should  be  calculated  as  indi- 
cated on  p.  80. 

Questions. 

1.  How  might  the  present  method  be  adapted  to  find  the  specific  heat  of 
a  liquid? 

2.  Considering  evaporation,  loss  of   heat  when  transferring  the  hot  body, 
and  any  other  sources  of  error  that  may  occur  to  you,  is  your  result  more  prob- 
ably too  high  or  too  low? 

XX.    RATIO  OF  SPECIFIC  HEATS  OF  GASES. 
(Clement  and  Desormes'  Method.) 

References — Elementary:  Duff,  §§287,  343-345;  Ames,  pp.  252-256;  Kimball, 
§404;  Reed  &  Guthe,  §175;  Watson,  §§259-260. — More  Advanced:  Edser 
(Heat],  pp.  317-325;  Poynting  &  Thomson  (Heat),  pp.  288-294. 

The  gas  is  compressed  into  a  vessel  until  the  pressure  has  a 
value  which  we  will  designate  by  pi.  The  vessel  is  then  opened 
for  an  instant,  and  the  gas  rushes  out  until  the  pressure  inside 
falls  to  the  atmospheric  pressure,  p0.  This  expansion  may  be 
made  so  sudden  that  it  is  practically  adiabatic  and  the  tempera- 
ture of  the  gas  will  therefore  fall.  After  the  vessel  has  been 
closed  for  a  few  minutes,  the  gas  will  have  warmed  to  the  room 
temperature,  /,  and  the  pressure,  p%,  will  be  above  that  of  the 
atmosphere.  Consider  one  gram  of  the  gas.  During  the 
adiabatic  expansion,  its  volume  changed  from  v\  to  vz,  according 
to  the  adiabatic  equation  for  pressure  and  volume  (see  references) 


( 


po 


Since  the  initial  and  final  temperatures  are  the  same,  and  since 
the  volume  remains  Awhile  the  gas  is  warming  and  the  pressure 
is  rising  from  po  to  p%,  by  Boyle's  Law 


Pi. 

po 


RATIO   OF   SPECIFIC   HEATS   OF   GASES.  85 

Hence  7,  the  ratio  of  specific  heats,  is  given  by  the  equation 


A  large  carboy  is  mounted  in  a  wooden  case  and  may  be  sur- 
rounded with  cotton  batting.  The  neck  is  closed  with  a  rubber 
stopper  through  which  passes  a  T-tube  connected  on  one  side 
with  a  compression  pump  (e.  g.,  a  bicycle  pump),  and  on  the 
other  side  with  a  manometer  containing  castor  oil.*  A  large 


FIG.  24. 

glass  tube,  which  may  be  closed  by  a  rubber  stopper,  also  passes 
through  this  large  stopper.  A  little  sulphuric  acid  in  the  bottom 
of  the  carboy  keeps  the  air  dry.  A  very  fine  copper  wire  and  a 
very  fine  constantin  wiref  pass  tightly  through  minute  holes  in  the 

*  The  density  of  castor  oil  is  about  .97,  but  it  should  properly  be  determined 
(Exp.  IV). 

f  No.  40  "Advance"  wire,  Driver-Harris  Co.,  Harrison,  N.  J. 


86  HEAT. 

stopper  and  meet  at  the  center  of  the  carboy,  in  a  minute  drop 
of  solder. 

The  air  in  the  carboy  is  compressed  until  the  difference  in 
pressure  is  about  40  cm.  of  oil  (  =  pi  —  po).  The  tube  connecting 
with  the  pump  is  closed,  and,  after  waiting  about  15  minutes  to 
allow  the  air  inside  to  regain  its  initial  temperature  (as  shown  by 
the  pressure  becoming  constant),  the  ends  of  the  oil  column  are 
carefully  read.  During  the  first  part  of  the  time  the  flask  should 
be  uncovered  to  facilitate  cooling,  but  during  the  latter  part  the 
carboy  should  be  well  insulated  with  cotton  batting.  The  air 
inside  is  momentarily  allowed  to  return  to  atmospheric  pressure 
by  removing,  for  about  one  second,  the  rubber  stopper  from  the 
glass  tube.  After  waiting  until  the  air  inside  has  assumed  the 
room  temperature  (shown  by  the  pressure  becoming  constant), 
the  final  pressure  p2  is  determined.  The  cotton-wool  had  better 
be  removed  during  this  stage.  When  several  repetitions  of  the 
experiment  have  given  familiarity  with  the  apparatus  and  pro- 
cedure, determine  also  the  change  in  temperature. 

Connect  the  wires  to  a  galvanometer,  apply  the  initial  compres- 
sion pi,  and  observe  the  reading  of  the  galvanometer  when  it  has 
become  steady.  Remove  the  stopper  as  before  (for  not  over  one 
second),  replace  the  stopper,  and  observe  the  galvanometer 
reading.  The  proper  reading  to  record  is  the  fairly  steady  de- 
flection which  is  attained  immediately  after  the  stopper  is  re- 
moved. There  are  liable  to  be  rapid  fluctuations  which  should  be 
disregarded,  and  of  course  the  temperature  does  not  long  remain 
steady,  owing  to  heating  or  cooling  from  the  outside.  Record 
as  before  the  final  pressure  p*.  Repeat  several  times,  starting 
with  the  same  initial  pressure  pi.  Record  the  temperature  of  the 
room,  /,  and  pQ,  the  height  of  the  barometer  (p.  19). 

If  the  constants  of  the  galvanometer  are  not  known  the  gal- 
vanometer, together  with  the  thermo-couple  in  series,  should  be 
calibrated  as  described  on  page  170.  To  eliminate  residual 
thermo-electric  effects,  make  repeated  alternate  observations 
of  the  deflection  with  a  suitable  value  of  r  (Fig.  54)  to  give  a 
deflection  of  several  centimeters,  and  with  r  =  o.  Calculate  the 
constant  from  the  mean  difference  in  the  readings. 

Calculate  7,  the  ratio  of  specific  heats  by  the  above  equation. 


LATENT   HEAT   OF   FUSION.  87 

Calculate  the  change  of  temperature  from  the  mean  of  the  gal- 
vanometer deflections  and  the  constants  of  the  thermocouple  and 
galvanometer.  (A  copper-constantin  thermocouple  has  an 
e.  m.  f.  of  very  approximately  40  micro-volts  per  degree.) 

Compare  the  result  with  Ti  —  T0  where  Ti  is  /  +  273  and  T0 
is  calculated  from  the  adiabatic  equation  for  temperature  and 
pressure  (see  references). 


i/M-y-i 

o       \PQ/      7 


Even  though  exceedingly  fine  wire  is  employed  (see  note, 
p.  85),  the  heat  capacity  of  the  wire  is  relatively  so  great  that 
the  thermocouple  will  not  show  the  full  change  of  temperature. 

Draw  a  curve  with  volumes  as  abscissae,  and  pressures  as 
ordinates,  which  will  represent  the  changes  in  this  experiment. 

(Let  specific  volumes,  i.  e.,  volumes  of  one  gram,  be  abscissae.  Calculate 
from  Table  VI  and  the  laws  of  gases  the  specific  volumes  corresponding  to  the 
room  temperature  and  po,  pi,  and  fa,  and  draw  the  corresponding  isothermal. 
Draw  the  horizontal  line  corresponding  to  po.  Draw  a  vertical  through  the 
point  corresponding  to  fa  on  the  above  isothermal.  The  intersection  of  these 
two  straight  lines  will  evidently  be  po,  Vz.) 

Questions. 

1.  Explain  why  one  specific  heat  is  the  greater. 

2.  Did  the  initial  compression  take  place  isothermally?     Explain. 

3.  Do  you  see  any  objection  to  an  initial  exhaustion  of  the  gas  in  place  of 
the  compression? 

4.  What  are  the  advantages  and  disadvantages  of  a  large  opening?     Short 
time  of  opening?     Castor  oil  manometer? 

5.  How  would  an  aneroid  manometer  be  preferable  in  this  experiment  to  a 
liquid  manometer? 


XXI.   LATENT  HEAT  OF  FUSION. 

References — Elementary:  Duff,  §§302-303,  305-307;  Ames,  pp.  260-261;  Crew, 
§286;  Edser  (Heat),  pp.  145-149;  Kimball,  §426;  Reed  &  Guthe,  §194; 
Spinney,  §173;  Watson,  §211. — More  Advanced:  Poynting  &  Thomson 
(Heat),  pp.  204-208;  Watson  (Pr.),  §88. 

The  latent  heat  of  fusion  of  a  substance  is  the  number  of  calories 
required  to  melt  one  gram  of  the  substance.  The  most  common 
method  of  measuring  it  is  a  method  of  mixture  similar  to  that 
used  in  finding  the  specific  heat  of  a  solid.  A  known  mass  of  the 


88  HEAT. 

solid  at  its  melting-point  is  placed  in  a  known  mass  of  the  liquid 
at  a  known  temperature,  and  the  temperature  of  the  liquid  ob- 
served after  the  solid  has  completely  melted.  Allowance  must 
be  made  for  the  water  equivalent  of  the  calorimeter  and  correc- 
tion must  be  made  for  the  effect  of  radiation  to  or  from  the  calori- 
meter while  melting  is  taking  place.  The  error  due  to  radiation 
may  be  made  small  by  having  the  liquid  initially  as  much  above 
the  temperature  of  its  surroundings  as  finally  it  falls  below. 
Thus  loss  and  gain  by  radiation  will  approximately  balance. 
Nevertheless,  since  the  calorimeter  will  probably  not  be  the  same 
length  of  time  above  the  temperature  of  the  surroundings  as 
below,  there  will  be  a  residual  error  for  which  correction  must  be 
made. 

The  calorimeter  consists  as  usual  of  an  inner  can  polished  on 
the  outside  to  diminish  radiation,  and  enclosed  in  an  outer  can 
polished  on  the  inside.  The  space  between  the  two  cans  may  be 
filled  by  cotton-wool  to  prevent  air  currents,  and  still  further 
prevent  communication  of  heat.  The  inner  can  is  weighed, 
first  empty  and  then  half-filled  with  warm  water  about  10°  above 
the  room  temperature.  It  is  then  placed  in  the  outer  can  as 
described  above  and  covered  by  a  wooden  cover  having  holes 
for  thermometer  and  stirrer  and  a  hinged  cover  giving  access  to 
the  inner  vessel.  The  temperature  is  carefully  observed  and 
recorded  each  minute  for  five  minutes.  In  the  meantime,  ice 
is  broken  to  pieces  of  about  a  cubic  centimeter  in  volume.  These 
pieces  are  carefully  dried  in  filter  paper.  A  careful  observation 
of  the  temperature  of  the  water  in  the  calorimeter  having  been 
made  and  the  time  noted,  a  piece  of  ice  is  dropped  in  without 
splashing  and  kept  under  water  by  a  piece  of  wire  gauze  attached 
to  the  stirrer.  The  temperature  is  noted  every  half -minute  as 
the  ice  melts,  the  water  meantime  being  kept  stirred.  The  rate 
at  which  ice  is  dropped  in  is  regulated  simply  by  the  rate  at  which 
it  can  be  dried  and  the  temperature  and  time  noted. 

The  process  is  continued  until  the  temperature  has  fallen  to 
about  the  same  amount  below  room  temperature  as  it  was 
initially  above.  Then  the  addition  of  ice  is  discontinued  and  the 
temperature  of  the  water  further  observed  and  recorded  every 


LATENT  HEAT  OF  VAPORIZATION.  89 

minute  for  four  or  five  minutes.  Finally,  from  the  weight  of  the 
inner  can  and  its  contents  the  mass  of  the  ice  is  deduced,  and 
the  latent  heat  is  calculated.  After  the  proper  weight  of  ice  has 
been  ascertained,  the  experiment  should  be  repeated  with  a  single 
piece  of  approximately  this  weight.  As  there  are  considerable 
sources  of  error  that  cannot  be  eliminated,  the  whole  determina- 
tion should  be  repeated  as  often  as  time  will  permit. 

All  the  temperature  observations  should  be  plotted  against 
the  time  and  the  radiation  correction  determined  as  described 
on  pages  59  and  60.  In  reporting,  consider  the  possible  error  of 
your  result  so  far  as  it  depends  on  the  possible  error  of  your 
weighings  and  observations  of  temperature.  State  also  any  other 
sources  of  error  that  may  have  affected  your  result. 

Questions. 

1.  What  advantages  are  there  in  the  use  of  one  large  lump  over  an  equal 
mass  of  small  ones? 

2.  Why  must  the  water  in  the  inner  vessel  be  pure? 

3.  Is  it  preferable  to  have  the  air  about  the  calorimeter  moist  or  dry? 
Explain. 


XXII.   LATENT  HEAT  OF  VAPORIZATION. 

References — Elementary:  Duff,  §314;  Ames,  p.  269;  Crew,  §287;  Edser  (Heat}, 
pp.  150-159;  Kimball,  §442;  Reed  &  Guthe,  §208;  Spinney,  §174;  Watson, 
§214. — More  Advanced:  Poynting  &  Thomson  (Heat),  pp.  178-184. 

The  latent  heat  of  vaporization  of  a  substance  is  the  number  of 
calories  required  to  change  one  gram  of  the  substance  from  liquid 
to  vapor.  The  usual  method  of  measuring  it  is  a  method  of 
mixture.  A  known  mass  of  vapor,  at  a  known  temperature,  is 
discharged  into  a  known  mass  of  liquid,  at  a  known  initial 
temperature,  and  the  final  temperature  is  noted.  The  same 
precautions  are  necessary  as  in  finding  the  latent  heat  of  fusion. 
The  arrangement  of  the  calorimeter  is  also  the  same.  To 
minimize  radiation  the  water  should  be  initially  as  much  below 
room  temperature  as  it  finally  rises  above,  say  15°.  The  initial 
rate  of  warming  should  also  be  obtained,  and  also  the  final  rate 
of  coolirig. 

Several  different  forms  of  boiler  have  been  devised  for  the 
purposes  of  this  determination.  Two  will  be  briefly  described. 


HEAT. 


In  Berthelot's  boiler  the  delivery  tube  passes  out  through  the 
bottom  of  the  boiler,  which  is  heated  by  a  ring  burner  that  sur- 
rounds the  tube.  Thus  the  tube  is  so  far  as  possible  jacketed 
by  the  boiling  water.  The  usual  form  of  this  boiler  is  somewhat 
fragile,  but  a  good  substitute  may  be  made  from  a  round-bot- 
tomed boiling  flask  the  neck  of  which  has  been  shortened  (Fig.  25) . 
In  the  electrically  heated  boiler  the  heating  of  the  water  is  pro- 
duced by  a  coil  of  wire  that  is  immersed  in  the  water  and  is  heated 
by  a  current  of  electricity.  The  current  must  be  kept  regulated 

by  a  rheostat,  so  that  boiling  proceeds 
at  a  moderate  rate. 

The  chief  difficulty  is  in  delivering 
the  steam  dry.  Condensation  is  apt  to 
take  place  in  the  delivery  tube.  This 
can  be  reduced  by  inserting  a  trap  in 


FIG.  25. 


FIG.  26. 


the  delivery  tube  between  the  boiler  and  the  calorimeter.  The 
trap  should,  from  time  to  time,  be  cautiously  heated  by  a  Bunsen 
burner  to  prevent  condensation,  but  in  general,  it  is  better  to  dis- 
pense with  the  trap  and  make  the  exposed  part  of  the  delivery 
tube  as  short  as  possible  and  carefully  cover  it  with  cotton-wool. 
If  the  delivery  tube  simply  passed  to  a  sufficient  depth  beneath 
the  water,  the  steam  would  be  delivered  at  greater  than  at- 
mospheric pressure,  as  the  pressure  of  a  certain  depth  of  water 
would  have  to  be  overcome.  Hence  it  is  better  to  let  the  de- 
livery tube  pass  into  a  condensing-box  immersed  in  the  water. 
The  latter  must  also  be  open  to  the  atmosphere  by  another  tube. 


LATENT   HEAT   OF   VAPORIZATION.  9 1 

To  prevent  any  escape  of  steam  by  this  tube  it  may  be  closed  by  a 
little  cotton-wool.  The  amount  of  steam  that  has  been  con- 
densed is  obtained  by  weighing  the  condensing-box  (well  dried) 
before  it  is  placed  in  the  calorimeter,  and  again  with  the  contained 
water  at  the  end  of  the  experiment.  The  temperature  of  the 
steam  is  deduced  from  the  barometric  pressure.  A  pressure 
gauge  attached  to  the  boiler  affords  a  means  of  estimating  how  far 
the  pressure  differs  from  atmospheric  pressure. 

For  the  best  results,  certain  precautions  must  be  observed. 
The  delivery  tube  must  not  be  connected  to  the  condensing-box 
until  steam  has  begun  to  pass  freely,  and  as  dry  as  possible,  from 
the  tube.  Connection  should  not  be  attempted  until  care  has 
been  taken  that  everything  is  ready  for  making  a  deft  and  prompt 
connection.  For  five  minutes  before  the  connection  is  made  the 
water  should  be  well  stirred  and  the  temperature  observed  and 
recorded  each  minute.  After  connection  the  temperature  is 
read  every  half-minute,  the  water  meantime  being  kept  well 
stirred  by  a  stirrer  (which  should  be  of  the  same  material  as  the 
calorimeter  and  condenser  in  order  to  simplify  the  calculation  of 
the  water  equivalent).  The  flame  of  the  ring-burner  must  be 
regulated  so  that  the  steam  does  not  pass  too  rapidly.  This 
may  be  gauged  by  the  rate  of  the  rise  of  the  temperature  of  the 
calorimeter,  which  should  not  exceed  4°  or  5°  per  minute.  In 
finding  the  subsequent  rate  of  cooling,  the  boiler  should  be  dis- 
connected from  the  condenser,  the  tube  leading  to  the  condenser 
should  be  closed  by  plugs  of  cotton-wool  to  prevent  evaporation 
and  the  temperature  observed  and  recorded  every  minute  for 
five  minutes;  but  in  subsequently  weighing  the  condenser  the 
wool  should  not  be  included.  The  whole  determination  should 
be  repeated  as  many  times  as  possible. 

A  formula  for  the  calculation  of  the  latent  heat  may  be  readily 
worked  out.  Account  must  be  taken  of  the  water  equivalent 
of  calorimeter,  condenser,  and  stirrer.  The  correction  for  radia- 
tion is  made  by  the  method  stated  on  pages  59-61. 

The  possible  error  of  the  result,  so  far  as  it  depends  on  the  read- 
ings made,  should  be  calculated,  and  other  possible  sources  of 
error  should  be  mentioned. 


92  HEAT. 

Questions. 

1.  State  the  advantages  and  disadvantages  of  a  rapid  flow  of  steam. 

2.  Explain  why  the  latent  heat  should  vary  with  the  atmospheric  pressure. 

3.  Must  the  boiling  water  be  pure?     Explain. 


XXIII.   LATENT  HEAT  OF  VAPORIZATION. 
Continuous-flow  Method. 

References — Elementary:  Duff,  §§285,  314;  Ames,  p.  269;  Crew,  §287;  Edser 
(Heat),  pp.  150-159;  Kimball,  §442;  Reed  &  Guthe,  §208;  Spinney,  §174; 
Watson,  §214. — More  Advanced:  Poynting  Of  Thomson  (Heat),  pp.  178- 
184. 

The  following  continuous-flow  method  will  give  far  more 
accurate  results  than  the  method  of  mixture  of  the  preceding 
experiment. 

The  apparatus  for  this  method  may  be  readily  constructed 
from  a  Liebig's  condenser.  Water  enters  at  D  and  leaves  at  C 
through  T-tubes  connected  to  the  condenser  by  short  rubber 
tubes.  Superheated  steam  enters  at  A  through  a  T-tube  and 
the  condensed  water  drops  into  a  covered  beaker  E.  The  steam 
is  superheated  as  it  flows  through  a  glass  tube  FE.  This  is  first 
covered  with  asbestos  over  which  a  heating  coil  of  wire*  is 
wrapped,  the  coil  being  covered  by  a  second  layer  of  asbestos. 
AB  and  FE  are  mounted  on  a  wooden  frame  and  AB  is  thickly 
covered  with  cotton-wool  to  prevent  radiation.  Thermometers 
TI,  TZ,  T2,  ,r4,  give  the  respective  temperatures  of  the  super- 
heated steam,  the  outflowing  water,  the  inflowing  water,  and  the 
water  of  condensation.  The  supply  of  water  may  come  from  the 
water  mains,  if  this  is  sufficiently  constant  in  temperature.  It 
is,  however,  much  better  to  have  a  supply  of  from  5  to  10  gallons 
in  an  elevated  tank  and  keep  the  flow  constant  by  an  overflow 
regulator  as  indicated  in  Fig.  31  (Exp.  XXVIII). 

The  boiler  to  supply  the  steam  should  be  large  enough  to  allow 
of  a  flow  for  two  hours  without  refilling  (one  to  two  liters  will 
suffice) .  The  current  in  the  superheating  coil  should  be  regulated 
by  a  rheostat  so  that  the  superheated  steam  is  at  about  105°. 
Some  time  should  be  spent  in  testing  adjustments  to  obtain  a 

*"Nichrome"  wire  (supplied  by  the  Driver-Harris  Co.,  Harrison,  N.  J.) 
is  very  suitable. 


LATENT  HEAT  OF  VAPORIZATION. 


93 


suitable  current  and  a  rate  of  flow  of  water  that  will  give  a  rise  of 
temperature  of  about  20°.  The  tank  should  be  connected  with 
the  water  service  so  that  it  can  be  readily  filled.  The  water  as 
it  comes  from  the  mains  will  probably  be  below  room  tempera- 
ture and  this  is  an  advantage,  since  with  a  suitable  rate  of  flow 


FIG.  27. 

of  the  steam  the  water  that  drops  into  E  will  differ  but  little 
from  room  temperature  and  will  suffer  little  loss  of  heat  by 
radiation.  This  will  require  a  proper  regulation  of  the  burner 
that  heats  the  boiler.  The  burner  should  be  surrounded  by  a 
shield  of  sheet-iron  or  asbestos  to  prevent  fluctuations  caused  by 
air-currents. 


94  HEAT. 

The  thermometers  TI,  T2,  and  T3  should  be  read  once  a  minute 
(e.  g.,  T2  20  sec.  after  TI  and  T3  20  sec.  after  7"2).  From  the 
mean  of  each  of  these  readings,  the  temperature  of  E,  the  mass 
of  water  that  flows  out  at  C,  and  the  mass  of  the  water  that 
drops  into  E,  the  latent  heat  can  be  calculated.  The  specific 
heat  of  the  superheated  steam  may  be  taken  as  0.5.  A  formula 
can  be  readily  constructed  to  express  the  fact  that  the  heat  given 
up  by  the  steam  and  condensed  water  equals  the  heat  carried  off 
by  the  current  of  water. 

Questions. 

1 .  Why  does  not  the  water  equivalent  of  the  condenser  need  to  be  considered  ? 

2.  How  could  you  find  the  amount  of  error  due  to  conduction  of  heat  from 
the  superheater  to  the  water  in  the  condenser? 

3.  How  could  you  find  the  amount  of  error  due  to  radiation  from  the  con- 
denser? 

4.  What  other  sources  of  possible  error  are  there  in  this  method? 

5.  If  two  very  accurate  thermometers  were  supplied  and  two  less  accurate, 
how  should  they  be  distributed  to  produce  the  least  error  in  the  result? 


XXIV.    THERMAL  CONDUCTIVITY. 

References — Elementary:  Duff,  §§326-327;  Ames,  p.  288;  Crew,  §§254,  256; 
Edser  (Heat),  pp.  416-430;  Kimball,  §420;  Reed  &  Guthe,  §§222-223; 
Spinney,  §§207-209;  Watson,  §§106-107. — More  Advanced:  Poynting  & 
Thomson  (Heat),  pp.  88-109;  Watson  (Pr.),  §§106-107. 

The  thermal  conductivity  of  a  substance  is  the  amount  of  heat 
transmitted  per  second  per  unit  of  area  through  a  plate  of  the 
substance  of  unit  thickness,  the  temperature  of  the  two  sides 
differing  by  i°  and  the  flow  having  become  steady.  If  K  be 
the  thermal  conductivity,  and  if  a  plate  of  thickness  /  and  area 
A  be  kept  with  one  side  at  a  temperature  t,  and  the  other  at  a 
lower  temperature,  t',  the  number  of  calories  that  will  flow 
through  the  plate  in  time  T,  after  the  flow  has  become  steady, 
will  be 

KA(t-f)T 


Q= 


l 


whence  K  can  be  derived  if  the  other  quantities  are  observed 
or  measured. 

Thermal  conductivity  is  in  general  difficult  to  measure  satis- 
factorily.    The  following  very  simple  method  cannot  be  relied 


THERMAL   CONDUCTIVITY. 


95 


on  to  closer  than  a  few  per  cent.,  but  it  only  requires  a  small 
portion  of  the  time  that  the  more  accurate  methods  call  for. 

A  rod  or  wire  of  the  substance  to  be  tested  is  inserted  at  one 
end  into  a  heavy  block  of  metal,  which  is  heated  to  a  constant 
high  temperature  in  a  bath,  through  the  bottom  of  which  the 
rod  passes.  At  its  lower  end  the  rod  is  screwed  into  a  heavy 
block  of  brass  or  copper  of  mass  M  and  specific  heat  s,  which  is 
initially  at  a  very  low  temperature.  Heat  is  thus  conducted 
by  the  rod  from  the  bath  to  the  lower  block.  If  the  latter 
neither  lost  nor  gained  heat  by 
convection  or  radiation,  and  if 
there  were  no  losses  from  the  sides 
of  the  rod,  we  could  calculate  the 
conductivity  of  the  rod  from  its 
dimensions  and  the  mass,  specific 
heat,  and  rise  of  temperature  of 
the  lower  block.  The  loss  of 
heat  from  the  surface  of  the 
rod  is  almost  wholly  prevented 
by  enclosing  it  by  a  glass  tube, 
which  does  not  come  into  direct 
contact  with  the  rod,  and  wrapping  the  glass  tube  with  cotton 
wool  and  paper. 

To  allow  for  radiation  or  convection  to  or  from  the  lower  block 
the  experiment  is  modified  as  follows :  The  block  is  enclosed  in  a 
vessel  surrounded  by  a  water-jacket,  through  which  water  at  a 
constant  temperature,  /',  circulates.  Now,  the  rate  at  which  the 
lower  block  receives  heat  through  the  rod,  when  the  former  is 
at  the  temperature  /',  is  the  mean  of  the  rates  at  which  it  receives 
heat  when  it  is  ir  degrees  below  /',  and  when  it  is  TT  degrees  above 
/'.  For  let  jR,  RI,  and  R2  represent  the  rates  of  conduction  of  heat 
(flow  of  heat  in  one  second)  to  the  lower  block  at  temperatures  /', 
/'  —  TT,  and  t'  -f  TT,  the  upper  end  being  at  temperature  t.  Then 

KA(t-t') 

K  Alt -(?-*)] 

Rl~      ~~r 


I 


FIG.  28. 


96  HEAT. 

whence 


Again,  when  the  lower  block  is  at  the  same  temperature  as  the 
jacket,  it  neither  receives  heat  from  nor  gives  heat  to  the  jacket. 
And  when  it  is  TT  degrees  below  it  gains  heat  as  rapidly  as  it  loses 
heat  when  it  is  TT  degrees  above.  Thus  by  taking  the  mean  rate 
as  above,  the  effects  of  radiation  to  or  from  the  block  are  elimi- 
nated. In  fact,  adding  a  to  RI,  to  allow  for  the  gain  by  radiation, 
and  subtracting  a  from  jR2,  to  allow  for  loss  by  radiation,  would 
leave  RI  +  R%  unchanged. 

The  same  would  hold  true  for  any  other  pair  of  temperatures 
equidistant  from  the  temperature  of  the  jacket.  If  the  rates  of 
rise  at  two  temperatures  equidistant  from  the  temperature  of  the 
jacket  be  r\  and  r%,  by  what  has  been  said  the  rate  at  the  tempera- 
ture of  the  jacket  would  be  J/2  (/i  +  r%).  Hence,  the  rate  at 
which  the  body  must  be  gaining  heat  is  Ms1/^  (r\  +  r2).  Hence, 
by  the  definition  of  thermal  conductivity, 

KA(t-t') 
' 


or, 

K 


A(t-t') 


The  lower  block  should  be  cooled  initially  to  about  12°  below 
the  temperature  of  the  water  that  circulates  through  the  jacket 
by  being  placed  in  a  bath  of  ice  and  water' (or  snow).  When 
taken  out,  it  must  be  carefully  dried.  The  jacket  may  be  kept 
at  a  constant  temperature  by  water  passing  and  repassing 
through  it  between  two  large  vessels,  which  are  alternately  raised 
and  lowered  about  every  five  or  ten  minutes.  The  temperature 
of  the  water  should  be  frequently  read  by  a  thermometer  (which 
may  conveniently  pass  through  a  large  cork  that  floats  on  the 
surface  of  the  water).  If  the  temperature  of  the  water  should 
show  a  tendency  to  rise  or  fall,  a  snrall  quantity  of  cooler  or 
warmer  water,  respectively,  may  be  added.  If  the  vessels  be 
large  and  the  temperature  of  the  room  does  not  vary  widely, 


THE  MECHANICAL  EQUIVALENT  OF  HEAT.          97 

there  should  be  no  difficulty  in  keeping  the  water  constant  to 
within  .2°  for  a  sufficient  length  of  time. 

The  hot  bath  is  in  the  form  of  a  trough,  which  is  heated  at 
one  end,  while  the  conducting  rod  passes  into  the  tank  at  the 
other  end.  To  prevent  direct  radiation  from  the  burner  to  the 
rod,  thick  screens  of  wood  and  asbestos  are  interposed.  The 
temperature  of  the  lower  block  should  be  read  at  least  every 
minute  by  means  of  a  thermometer  passing  through  the  cork  or 
fiber  cover  and  inserted  into  a  hole  in  the  block,  the  unoccupied 
space  in  the  hole  being  filled  with  mercury.  The  temperature  of 
the  upper  block  should  be  read  frequently  by  a  thermometer 
thrust  into  it.  Small  quantities  of  boiling  water  should  be  added 
frequently  to  the  bath  to  compensate  for  evaporation. 

The  readings  of  the  temperature  of  the  lower  block  are  likely 
to  be  somewhat  irregular.  They  should  be  plotted  against  the 
time  and  a  smooth  curve  drawn  as  described  on  pages  10  and  n. 
From  this  curve  the  mean  rate  of  rise  for  each  pair  of  degrees 
equidistant  from  the  temperature  of  the  jacket  should  be  ob- 
tained, and  the  final  mean  of  all  taken  in  calculating.  It  is, 
however,  to  be  noted  that  the  temperatures  nearest  the  jacket 
temperature  should  give  the  best  results,  since  there  the  radiation 
is  a  minimum,  and  therefore  any  defect  in  the  method  of  correct- 
ing for  radiation  a  minimum. 

(For  comparing  the  conductivities  of  poorly  conducting  sub- 
stances Lees  and  Chorlton's  apparatus  is  quite  satisfactory. 
Directions  for  its  construction  and  manipulation  are  given  in 
Robson's  Heat,  page  135.) 

XXV.    THE  MECHANICAL  EQUIVALENT  OF  HEAT. 

References — Elementary:  Duff,  §§290-291;  Ames,  pp.  203-205;  Crew,  §289; 
Edser  (Heat},  Chap.  XII;  Kimball,  §410;  Reed  &  Guthe,  §177;  Spinney, 
§218;  Watson,  §§250-251. — More  Advanced:  Griffith,  Thermal  Measurement 
of  Energy,  Chap.  Ill;  Poynting  &  Thomson  (Heat),  pp.  116-128;  Rowland, 
Physical  Papers,  pp.  343-476. 

The  mechanical  equivalent  of  heat  is  the  number  of  units  of 
mechanical  energy  that,  completely  turned  into  heat,  will  pro- 
duce one  unit  of  heat,  or,  in  the  c.  g.  s.  system,  the  number  of 
ergs  in  a  calorie.  The  apparatus  here  described  is  a  copy  of  that 

7 


98 


HEAT. 


used  in  the  University  of  Cambridge,  England,  and  the  following 
description  and  introduction  is  partly  taken  from  that  issued  to 
students  in  that  university.  In  this  apparatus  mechanical 
energy  is  expended  in  working  against  friction,  thus  producing 
heat,  which  is  measured  by  the  rise  in  temperature  of  a  known 
mass  of  water. 

A  vertical  spindle  .carries  at  its  upper  end  a  brass  cup.  Into 
an  ebonite  ring  concentric  with  the  cup  there  fits  tightly  one  of  a 
pair  of  hollow  truncated  cones.  The  second  cone  fits  into  the 
first,  and  is  provided  with  a  pair  of  steel  pins  which  correspond 


a 


CD 


i°L 


FIG.  29. 


to  two  holes  in  a  grooved  wooden  disk,  which  prevents  the  inner 
from  revolving  when  the  spindle  and  the  outer  cone  revolve. 
A  cast-iron  ring,  resting  on  the  disk  and  fixed  by  two  pins,  serves 
to  give  a  suitable  pressure  between  the  cones.  A  brass  wheel 
is  fixed  to  the  spindle,  and,  by  a  string  passing  round  this  wheel 
and  also  round  a  hand-wheel,  motion  is  imparted  to  the  spindle. 
A  pair  of  guide  pulleys  prevents  the  string  from  running  off  the 
wheel.  Above  the  wheel  is  a  screw  cut  upon  the  spindle.  This 
screw  actuates  a  cog-wheel  of  100  teeth,  which  makes  one  revolu- 
tion for  every  100  revolutions  of  the  spindle. 


THE  MECHANICAL  EQUIVALENT  OF  HEAT.          99 

To  the  base  of  the  apparatus  one  end  of  a  bent  steel  rod  is 
attached;  the  rod  can  be  fixed  in  any  position  by  a  nut  beneath 
the  base.  The  other  end  of  the  rod  carries  a  cradle,  in  which 
runs  a  small  guide  pulley  on  the  same  level  with  the  disk.  The 
cradle  turns  freely  about  a  vertical  axis.  A  fine  string  is  fastened 
to  the  disk  and  passes  along  the  groove  in  its  edge;  it  then  passes 
over  the  pulley  and  is  fastened  to  a  mass  of  200  or  300  grams. 
On  turning  the  hand-wheel  it  is  easy  to  regulate  the  speed  so 
that  the  friction  between  the  cones  just  causes  the  mass  to  be 
supported  at  a  nearly  constant  level.  To  prevent  the  string  from 
running  off  the  guide  pulley,  a  stiff  wire  with  an  eye  is  fixed  to  the 
cradle  and  the  string  is  passed  through  this  eye.  It  also  passes 
through  an  eye  fixed  to  the  steel  rod,  to  prevent  the  weight  from 
being  wround  up  over  the  pulley. 

The  rubbing  surfaces  of  the  cones  must  be  carefully  cleaned 
and  then  four  or  five  drops  of  oil  must  be  put  between  them ;  the 
bearings  of  the  spindle  and  guide  pulley  should  also  be  oiled. 
The  cones  are  then  weighed  together  with  the  stirrer.  The 
inner  cone  is  then  filled  to  about  I  cm.  from  its  edge  with  water 
5°  or  6°  below  the  temperature  of  the  room  and  the  system  is 
again  weighed.  The  cones  are  then  placed  in  position  in  the 
machine  and  a  thermometer  (divided  to  tenths  of  a  degree)  is 
hung  from  a  support  so  that  it  passes  through  the  central  aperture 
in  the  disk  and  almost  touches  the  bottom  of  the  inner  cone. 

One  observer,  X,  takes  his  place  at  the  hand-wheel,  and  the 
other,  Y,  at  the  friction  machine.  By  working  the  machine  the 
water  is  now  warmed  up  until  its  temperature  is  three  or  four 
degrees  below  that  of  the  room.  The  turning  is  then  stopped, 
the  index  of  the  counting-wheel  is  read,  and  the  temperature  of  the 
water  is  carefully  observed  every  minute  for  five  minutes.  Im- 
mediately after  the  last  reading,  X  turns  the  wheel  fast  enough 
to  raise  the  mass  until  the  string  is  tangential  to  the  edge  of  the 
disk.  If  the  string  be  not  tangential  the  moment  of  its  tension 
about  the  axis  of  revolution  is  seriously  diminished.  Y  stirs 
the  water  and  notes  the  temperature  at  each  passage  of  the  zero 
of  the  counting-wheel  past  the  index;  each  passage  of  the  zero 
after  the  first  corresponds  to  100  revolutions  of  the  spindle.  Y 
gives  a  signal  at  each  passage  of  the  zero  and  X  notes  the  time  by 


100  HEAT. 

aid  of  a  watch.  After  Y  has  recorded  the  temperature  upon  a 
sheet  of  paper  previously  ruled  for  the  purpose,  he  also  records 
the  time  observed  by  X.  When  the  temperature  has  risen  about 
as  high  above  room  temperature  as  it  was  initially  below,  the 
motion  is  stopped  and  the  readings  of  the  index  of  the  counting- 
wheel  and  of  the  thermometer  are  recorded.  Observations  of 
the  temperature  are  continued  every  minute  for  five  minutes, 
the  stirring  of  the  water  being  continued. 

The  temperature  observations  are  plotted  against  the  time, 
and  the  radiation  correction  is  determined  as  explained  on  pages 
59  and  60.  The  heat  produced  is  readily  calculated  from  the 
mass  of  water,  the  water  equivalent  of  the  cones  and  stirrer, 
and  the  corrected  rise  of  temperature. 

From  the  initial  and  final  readings  of  the  counting-wheel  and 
the  number  of  complete  revolutions  the  exact  number  (n)  of 
revolutions  made  by  the  spindle  is  deduced.  The  work  done  is 
calculated  as  follows:  When  the  spindle  has  made  n  turns  the 
work  spent  in  overcoming  the  friction  between  the  cones  is  the 
same  as  would  have  been  spent  if  the  outer  cone  had  been  fixed 
and  the  inner  one  had  been  made  to  revolve  by  the  descent  of  the 
mass  of  M  grams.  In  the  latter  case  M  would  have  fallen  through 
2irnr  cm.  where  r  is  the  radius  of  the  groove  of  the  wooden  disk, 
which  must  be  measured.  Hence  the  total  work  spent  against 
friction  and  turned  into  heat  is  2irnrMg  ergs.  In  the  repoit, 
estimate  the  possible  error  of  the  result  as  far  as  it  depends  upon 
the  errors  of  observations  and  measurements. 

Questions. 

1.  What  amount  of  error  is  due  to  neglect  of  the  work  spent  against  friction 
of  the  bearing  of  the  outer  cone? 

2.  Why  must  the  wheel  be  turned  faster  as  the  experiment  proceeds? 

3.  What  effect  on  the  result  has  the  variation  of  the  viscosity  of  the  oil? 

4.  Is  the  heat  or  the  work  measured  the  more  accurately?     Explain. 

XXVI.    THE  MELTING-POINT  OF  AN  ALLOY. 

References — Elementary:  Duff,  §304. — More  Advanced:  Ewell,  Physical  Chem- 
istry, pp.  271-272;  Findlay,  Phase  Rule,  pp.  220-223. 

If  an  alloy  is  melted  and  is  allowed  to  cool  while  its  tempera- 
ture is  continuously  observed,  and  a  curve  be  then  drawn  with 
times  as  abscissae  and  temperature  as  ordinates,  it  will  be  found 
that  at  certain  points  the  curvature  abruptly  changes,  the  fall 


THE   MELTING-POINT   OF    AN   ALLQY: 


of  temperature  being  decreased  or  even  ceasing.  At  the  moment 
corresponding  to  such  a  point,  the  alloy  is  radiating  heat  to  the 
room,  and  the  fact  that  its  temperature  does  not  fall  as  rapidly 
indicates  that  heat  is  being  produced  internally  by  some  change 
of  state  of  the  material.  Such  a  point  is,  therefore,  a  solidify  ing- 
point  of  some  constituent  of  the  alloy  or  of  the  eutectic  alloy. 

The  assigned*  metals  are  carefully  weighed  and  melted  in  an 
iron  cup.  A  copper-constantinf  thermocouple  is  plunged  into 
the  liquid  metal  and  kept  there  until  the  entire  mass  is  solid. 
A  porcelain  tube  should  cover  one  wire  for  some  distance  from  the 
junction.  The  terminals  are  connected  to  a  calibrated  galvano- 
meter through  a  resistance  such  that  the  maximum  deflection 
will  keep  on  the  scale.  The  galvanometer  is  read  every  half- 
minute  and  the  time  of  each  reading  is  noted.  When  the  read- 
ings are  commenced,  the  metal  should  be  considerably  above  the 
melting-point  and  the  readings  should  be  continued  for  some  time 
after  the  metal  is  apparently  solid.  The  galvanometer,  with 
thermocouple  and  series  resistance,  should  be  calibrated  as  in 
Exp.  XX. 

Plot  the  galvanometer  deflections  against  the  time.  De- 
termine the  electromotive  force  corresponding  to  the  galvano- 
meter deflections  where  the  curvature  changed,  and  from  the 
constants  of  the  thermocouple,  or  a  chart  giving  the  temperature 
for  different  electromotive  forces,  determine  the  temperature  of 
these  points.  Remember  that  the  thermocouple  merely  gives 
the  difference  in  temperature  between  the  two  junctions. 

Tabulate  the  observed  temperatures  of  these  transition  points 
and  your  opinion  of  what  they  represent. 

Questions. 

1.  Explain  why  the  second  transition  point  for  the  alloy  is  represented  by 
a  horizontal  portion  of  the  cooling  curve  while  the  first  transition   point  is 
merely  represented  by  a  change  of  curvature. 

2.  Will  the  temperature  of  the  first  point  vary  with  the  initial  concentration? 

3.  Will  the  temperature  of  the  second  transition  point  vary  with  the  initial 
concentration? 

4.  Define  "Eutectic  alloy." 

5.  Outline  a  methocl  for  determining  the  latent  heat  of  fusion  from  the 
cooling  curve. 

*  Tin  and  lead  are  suitable  metals.  The  changes  of  curvature  for  their 
alloy  are  more  distinct  if  the  former  is  in  excess.  The  eutectic  of  tin  and  lead 
is  composed  of  37%  lead,  63%  tin,  and  melts  at  182.5°  (Rosenhain  and  Tucker, 
Roy.  Soc.  Phil.  Trans.,  1908,  A.  209,  p.  89). 

f  "Advance"  Wire,  Driver-Harris  Co.,  Harrison,  N.  J. 


10:?  HEAT. 


XXVII.    HEAT  VALUE  OF  A  SOLID. 

References — Elementary:  Duff,  §289. — More  Advanced:  Ferry  and  Jones,  pp. 

237-242. 

(A)    HEMPEL  BOMB  CALORIMETER  (Constant-volume 
Calorimeter) 

A  pellet  of  the  fuel  to  be  tested  is  formed  in  a  press,  a  cotton 
cord  being  imbedded  with  a  loose  end.  After  being  pared  down 
to  about  I  gm.  and  brushed,  it  is  carefully  weighed.  It  is  then 
suspended  in  a  Hempel  combustion  bomb,  and  the  thread  is 
wrapped  around  a  platinum  wire  connecting  the  platinum  sup- 
ports of  the  basket.  The  terminals  attached  to  these  supports 
are  connected  with  several  Edison  or  storage  cells  sufficient  to 
just  bring  the  wire  to  a  brilliant  incandescence  (as  ascertained  by 
a  preliminary  trial). 

The  bomb  is  charged  with  oxygen  under  at  least  fifteen  at- 
mospheres' pressure,  either  from  a  charged  cylinder  or  produced 
by  a  retort.  Bomb  and  pressure  gauge  should  be  immersed  in 
water  while  the  oxygen  is  being  supplied.  Ascertain  that  the 
bomb  value  is  open  and  that  all  connections  are  screwed  tight. 
Open  the  cylinder  valve  (if  a  cylinder  is  used)  until  the  pressure 
becomes  high,  and  then  close.  Lift  the  bomb  out  of  the  water, 
loosen  one  of  the  connections,  and  allow  the  mixture  of  air  and 
oxygen  to  escape;  then  tighten,  replace  in  water,  open  the 
cylinder  valve  again  until  the  pressure  becomes  high  (at  least 
fifteen  atmospheres) ;  close  both  the  cylinder  valve  and  that  of  the 
bomb,  and  finally  disconnect  and  dry  the  bomb.  (If  the  oxygen 
is  produced  in  a  retort,  partly  fill  the  latter  with  a  five  to  one 
mixture  of  potassium  chlorate  and  manganese  dioxide,  connect 
to  the  bomb  and  pressure  gauge, "and  heat  the  upper  part  slowly 
with  a  Bunsen  burner.) 

Attach  the  electrical  terminals,  place  the  bomb  in  the  special 
vessel  containing  about  a  liter  of  water,  adjust  the  Beckmann 
thermometer  to  read  about  i°  (see  p.  61),  stir  the  water  con- 
tinually, and  read  its  temperature  every  half-minute  for  five 
minutes,  estimating  to  tenths  of  the  smallest  graduation.  Close 


HEAT   VALUE   OF  A   SOLID.  1  03 

the  electric  switch;    after  a  few  'seconds  open  it  and  read  the 
temperature  of  the  continually  stirred  water  for  ten  minutes. 

Let  H  be  the  heat  value  of  the  fuel  and  m  the  mass  of  the  speci- 
men, M  the  mass  of  the  water,  e  the  water  equivalent  of  the  bomb, 
/i  the  initial  temperature,  and  /2  the  final  temperature  (both  cor- 
rected for  radiation,  see  p.  59.)  Then 

i)      , 

cal.  per  gm. 


m 

The  best  method  in  practice  to  determine  e  is  to  repeat  the 
determination,  using  salicylic  acid  as  fuel,  and  assuming  its  heat 
value  to  be  5300  calories  per  gram. 

(B)    ROSENHAIN'S  CALORIMETER  (Constant-pressure 

Calorimeter) 
Reference  —  Phil.  Mag.,  VI,  4,  p.  451. 

Instead  of  burning  the  fuel  in  a  fixed  volume  of  highly  com- 
pressed oxygen,  the  oxygen  is  supplied  continuously  at  only 
slightly  above  atmospheric  pressure. 

The  coal  is  pulverized  and  a  sample  is  compressed,  in  a  special 
screw  press,  into  a  pellet  weighing  about  one  gram.  This  is 
placed  on  a  porcelain  dish  which  rests  on  the  bottom  of  the 
inside  chamber.  The  ignition  wire  should  be  about  3  cm.  of 
No.  30  platinum  wire  and  the  external  terminals  should  be  con- 
nected to  storage-battery  terminals  through  a  key  and  a  resistance 
such  that  the  wire  will  glow  brightly.  A  gasometer  is  charged 
with  oxygen  from  a  cylinder  or  generated  from  "oxone"  and 
water.  The  action  of  the  different  valves  having  been  studied, 
the  apparatus  should  be  assembled,  the  upper  side  valve  (see 
Fig.  30)  being  closed  and  the  ball  valve  lowered.  Connect 
with  the  oxygen  supply  through  a  wash-bottle,  turn  on  a  moder- 
ate stream  of  oxygen,  and  pour  into  the  outer  vessel  a  measured 
volume  of  water,  at  the  room  temperature,  so  that  the  combustion 
chamber  is  just  covered. 

If  a  Beckmann  thermometer  (see  p.  61)  is  used,  adjust  to  read 
between  o°  and  i°  in  this  water.  The  bulb  should  be  supported 
on  a  level  with  the  center  of  the  combustion  chamber.  Read 
the  temperature  every  half-minute  for  five  minutes,  then  in- 


104 


HEAT. 


crease  the  oxygen  current,  and  carefully  read  the  temperature 
and  the  time,  close  the  key  and  ignite  the  pellet  with  the  hot 
platinum  wire,  and  immediately  remove  the  wire.  During  such 
operations  it  is  best  to  hold  the  inner  vessel  steady  by  grasp- 
ing the  oxygen  inlet  tube.  Keep  the  water  pressure  in  the 
gasometer  constant,  and  as  combustion  proceeds  increase  the 
flow  of  oxygen.  If  possible,  read  the  thermometer  every  half- 
minute. 

When  combustion  has  ceased,  move  the  hot  wire  about  to 
ignite  any  unconsumed  particles.  Keep  the  wire  hot  as  short 
a  time  as  possible  and  remove  it  immediately  from  any  combustion, 
otherwise  it  is  liable  to  be  melted. 

Finally,  immediately  after  combustion  ceases,  turn  off  the 
oxygen  supply,  open  the  upper  valve,  and  raise  the  ball  valve, 
allowing  the  water  to  enter  the  inner  chamber. 
Then  at  once  force  out  the  water  by  closing 
both  valves  and  turning  on  the  oxygen.  Re- 
cord the  highest  temperature  and  the  time 
and  the  temperature  every  half-minute  for 
five  minutes.  For  radiation  correction,  see 
p.  59,  and  for  formulae,  see  (A)  preceding. 

To  determine  e,  assemble  the  apparatus,  in- 
cluding the  Beckmann  thermometer,  and  pour 
in  1000  c.c.  of  water.  Read  every  minute, 
for  five  minutes,  with  0.1°  thermometers,  the 
temperature  of  this  water  and  also  of  500  c.c. 
of  water  which  has  been  heated  to  about  50°. 
After  noting  the  time,  pour  in  the  hot  water 
and  continue  readings  every  minute  for  five  minutes,  keeping 
the  mixture  well  stirred. 

Plot  these  observations,  draw  mean  straight  lines  through  the 
three  sets  of  temperature  readings,  and  prolong  the  lines  until 
they  intersect  the  vertical  line  corresponding  to  the  instant  of 
mixing.  From  these  three  temperatures  and  the  water  masses 
calculate  e. 

For  anthracite  coal  add  sugar  in  the  proportion  3:1.  (Heat 
of  combustion  of  sugar  =  3900  calories  per  gram.) 


FIG.  30. 


HEAT   VALUE   OF   A   GAS   OR  LIQUID. 


105 


Questions. 

1.  Calculate  the  heat  value  of  (a)  one  kilo  of  this  substance,  (b)  one  short 
ton  in  B.  T.  U.  per  lb.,  (c)  the  mechanical  energy  equivalent  to  the  latter. 

2.  What  error  would  be  caused  by  (a)  an  error  of  20  in  the  water  equivalent? 
(b)  allowing  a  current  of  5  amperes  to  flow  through  a  platinum  wire  of  2 
ohms'  resistance  for  5  seconds?     (c)  neglecting  the  radiation  correction? 

3.  How  many  liters  of  oxygen  would  be  required  to  consume  a  pellet  of 
carbon  of  the  same  mass  as  the  coal  pellet? 


XXVIII.    HEAT  VALUE  OF  A  GAS  OR  LIQUID. 

References — Duff,  §285;  Ferry  and  Jones,  pp.  243-246. 

The  heat  value  will  be  determined  with  Junker's  calorimeter. 

(A)  A  measured  volume  &  liters)  of  gas  under  an  observed 
pressure,  p,  is  burned  in  the  calorimeter,  and  the  rise  of  tempera- 
ture, from  /i°  to  £2°,  of  a  mass  of  M  gr.  of  water,  is  determined. 
The  flow  of  water  and  gas  is  so  regulated  that  the  burned  gas 
leaves  the  calorimeter  at  approxi- 
mately the  temperature  of  the  enter- 
ing gas,  and  there  should  be  a  differ- 
ence of  at  least  6°  in  the  temperature 
of  the  in-  and  outflowing  water.  Also, 
the  flow  of  water  must  be  sufficient  to 
furnish  a  constant  small  overflow  at 
the  supply  reservoir  (see  Fig.  31).  The 
burner  should  be  lighted  outside  the 
calorimeter.  When  the  temperatures 
indicated  on  the  various  thermometers 
have  become  constant,  note  the  gaso- 
meter reading,  and  immediately  collect 
in  graduates  the  heated  overflowing 
water, -and  also  the  water  condensed 
by  the  combustion  of  the  gas.  Let 

the  mass  of  the  latter  be  m  gr.,  and  its  temperature  tf .  Note  the 
temperatures  of  the  inflowing  and  outflowing  water  every  15  sec. 
until  two  or  three  liters  have  passed  through.  Then  immediately 
note  the  gasometer  reading  and  remove  the  graduates. 

Assuming  that  condensation  of  the  gas  occurs  at  100°,  the  heat 
liberated  is  m  [536  +  100  — 1'\.     If  H  represents  the  heat  value  of 


FIG  31. 


IO6  HEAT. 

the  gas  in  gram-calories  per  liter  and  v  the  volume,  reduced  to  o°, 
and  760  mm., 


(B)  To  determine  the  heat  value  of  a  liquid  fuel,  the  gas 
burner  is  replaced  by  a  suitable  lamp  which  is  attached  to  one 
arm  of  a  balance.  The  rate  at  which  the  liquid  is  consumed  is 
determined  from  the  weights  in  the  pan,  on  the  other  side,  at 
different  times.  It  is  best  to  make  the  weight  in  the  pan  slightly 
deficient  and  note  the  exact  time  when  the  balance  pointer  passes 
zero,  as  the  liquid  is  consumed.  Practically  complete  combus- 
tion is  obtained  with  a  "Primus"  burner,  supplied  from  a  reser- 
voir where  the  liquid  is  under  considerable  pressure.  With 
very  volatile  liquids,  the  opening  of  the  burner  must  be  large  and 
the  pre-heating  tubes  must  be  in  the  cooler  part  of  the  flame. 

Express  your  results  in  (a)  calories  per  liter  (b)  B.  T.  U.  per 
gallon  or  cubic  foot. 

Questions. 

1.  Is  the  heat  value  of  a  gas,  in  calories  per  gram,  definite?     Per  liter? 

2.  What  difference  would  there  be  in  the  result  if  all  the  water  vapor  escaped 
without  condensing? 

3.  Why  is  no  radiation  correction  necessary?     The  water  equivalent  un- 
determined? 

4.  What  factors  determine  the  amount  of  water  of  condensation? 


XXIX.  PYROMETRY. 

References — Elementary:  Duff,  §§269-271,  338-341;  Kimball,  §§378,  672; 
Spinney,  §§167,  211,  213,  289. — More  Advanced:  Bulletin,  Bureau  of 
Standards,  1,2,  pp.  189-255;  Burgess  &  Le  Chatelier,  Measurement  of  High 
Temperatures,  Chaps.  VII  &  VIII;  Edser  (Heat),  pp.  339-411;  Watson 
(Pr.},  §§208-210. 

This  exercise  is  a  study  of  the  methods  used  in  the  measure- 
ment of  very  high  temperatures. 

(A)  A  hollow  black  body  (an  electric  furnace)  is  to  be  heated 
and  the  temperature  of  its  interior  is  to  be  determined  by  means 
of  a  calibrated  thermocouple.  A  platinum  resistance  thermom- 
eter is  to  be  calibrated,  and  also  an  incandescent  lamp  is  to  be 
calibrated  for  use  as  an  optical  pyrometer. 


PYROMETRY. 


107 


Electric  Furnace. — The  electric  furnace  (see  Fig.  32)  consists 
of  a  thin  porcelain  cylinder  about  15  cm.  long  and  10  cm.  in 
diameter  upog  which  is  wound  about  5  m.  of  No.  22  "  Nichrome" 
wire,*  if  a  220- volt  supply  is  to  be  used.  Whatever  the  voltage, 
the  winding  must  be  such  as  to  consume  about  half  a  kilowatt. 
The  ends  of  the  cylinder  are  closed  by  porcelain  caps  with  proper 
apertures  and  the  whole  is  surrounded  by  many  layers  of  asbestos. 

Heating  and  cooling  must  be  very  gradual  so  that  the  thermo- 
couple and  platinum  thermometer  may  acquire  the  tempera- 
ture of  the  furnace.  The  highest  temperature  should  not  exceed 
1000°. 

Thermocouple. — A  platinum  and  platinum  +  10%  rhodium 
thermocouple  should  be  connected  to  a  galvanometer  through 


FIG.  32. 

a  key  and  such  a  resistance  as  will  keep  the  deflection  on  the  scale 
at  the  highest  temperature.  The  galvanometer  with  resistance 
should  be  calibrated  as  a  voltmeter  (p.  170).  The  chart  or 
table  accompanying  the  couple  gives  the  temperature  of  the  hot 
junction  when  the  electromotive  force  is  known.  (The  cool 
junction  should  be  in  ice  and  water.) 

Platinum  Resistance  Thermometer. — The  platinum  resistance 
thermometer  consists  of  a  coil  of  fine  platinum  wire  (for  example 
50  cm.  of  No.  30)  wound  on  a  porcelain  or  mica  frame  and  sur- 
rounded by  a  glazed  porcelain  tube.  The  coil  constitutes  one 
arm  of  a  Wheatstone's  bridge.  If  there  is  a  pair  of  dummy 
*  See  note  bottom  of  page  92. 


IO8  HEAT. 

leads  they  should  be  connected  to  an  adjoining  arm  (see  figure) 
and  compensate  for  the  heating  of  the  lead  wires.  A  suitable 
switch  connects  the  galvanometer  to  either  the  bridge  or  the  ther- 
mocouple. 

Optical  Pyrometer. — The  optical  pyrometer  consists  essentially 
of  a  lens  and  a  miniature  incandescent  lamp.  The  lens  focuses 
the  interior  of  the  enclosed  furnace  (an  ideal  black  body)  on  the 
filament  of  the  lamp.  The  current  through  the  filament  is  ad- 
justed until  the  tip  of  the  filament  is  invisible  against  the  image  of 
the  furnace.  When  this  is  true,  both  must  be  emitting  similar 
light,  and  therefore  they  must  be  at  approximately  the  same 
temperature.  A  small  eye-piece  aids  in  observing  the  tip  of  the 
filament.  The  incandescent  lamp  filament  is  to  be  calibrated; 
i.  e.,  the  current  necessary  to  heat  the  tip  of  the  filament  to 
different  temperatures  is  to  be  determined.  The  temperature 
of  the  filament  is  determined  by  finding  the  temperature  of  the 
furnace,  by  the  thermocouple,  when  the  two  have  the  same 
temperature.  (The  incandescent  lamp  circuit  contains  an 
ammeter  for  measuring  the  current,  which  is  omitted  from  Fig. 

32.) 

Observations. — While  the  furnace  is  slowly  heating,  and,  if  time 
permit,  also  while  it  is  slowly  cooling,  observe  and  tabulate  at 
frequent  intervals,  (i)  the  galvanometer  deflection  when  con- 
nected to  the  thermocouple,  the  resistance  in  the  circuit  being 
also  noted,  (2)  the  resistance  of  the  platinum  resistance  ther- 
mometer, (3)  the  current  through  the  filament.  The  three  ob- 
servations should  be  made  in  succession  and  the  time  of  each 
observation  recorded.  (4)  Calibrate  the  galvanometer  as  a 
voltmeter  (p.  170).  (5)  Determine  with  the  optical  pyrometer 
the  temperature  of  various  distant  luminous  bodies,  e.  g.,  a 
luminous  gas  flame.  An  image  of  the  hot  body  is  formed  upon 
the  filament  and  the  current  through  the  latter  is  adjusted  until 
the  two  are  indistinguishable.  The  temperature  is  obtained 
later  with  the  help  of  curve  (d)  below. 

In  reporting,  (a)  tabulate  the  observations  outlined  above. 
(b)  Plot  the  calibration  curve  of  the  thermocouple,  (c)  Plot 
and  sketch  on  one  sheet  three  smooth  curves  with  time  as 
abscissae  and  as  ordinates,  galvanometer  deflections,  resistances, 


PYROMETRY. 

and  current  through  filament,  respectively,  each  with  a  suitable 
scale.  Draw  about  ten  properly  spaced  vertical  lines,  the  inter- 
sections of  which  with  the  three  curves  will  give  simultaneous 
readings,  and  from  the  intersections  with  the  curves  of  gal- 
vanometer deflection,  calculate  the  temperature  of  the  interior 
of  the  furnace  corresponding  to  each  vertical  line,  (d)  Make  a 
third  plot  with  current  through  filament  as  abscissae  and  cor- 
responding temperature  as  ordinates,  using  the  simultaneous 
values  obtained  in  (c).  (e)  In  a  fourth  plot  make  resistance 
abscissae  and  temperature  ordinates,  using  again  the  simultaneous 
values  obtained  in  (c).  Obtain  from  the  instructor  the  values  of 
the  resistance  at  o°  and  100°  and  plot  as  a  second  curve  the 
platinum  temperature  as  given  by  Callender's  equation 


pt  =  ioo  V~  ITS- 

-ft-ioo — -t\o 

For  a  number  of  points  such  as  200°,  400°,  etc.,  find  the 
difference  between  the  platinum  temperature,  pt,  and  the  true 
temperature,  t,  by  Callender's  correction  equation 


100 

Apply  this  correction  to  the  curve  last  drawn  and  compare  the 
corrected  curve  with  the  first,  experimental  curve.  Finally,  ob- 
tain from  (5)  and  (d)  the  temperature  of  the  objects  examined. 

(B)  If  a  platinum- wound  furnace  is  available,  the  temperature 
range  may  be  largely  increased  and  a  more  extensive  calibration 
curve  obtained  for  the  optical  pyrometer.  The  platinum  re- 
sistance thermometer  should,  however,  be  removed,  as  it  will  be 
injured  if  heated  above  1000°  C.  The  experiment  will  then  con- 
sist of  observations  of  the  galvanometer  deflection  due  to  the 
thermocouple  and  (after  the  interior  of  the  furnace  has  become 
luminous)  reading  of  the  current  through  the  filament  and,  finally, 
use  of  the  filament  for  determining  the  temperature  of  various 
luminous  bodies.  Calculation  and  report  will  be  similar  to  that 
outlined  above  with  the  omission  of  portions  referring  to  the 
platinum  resistance  thermometer. 


IIO  HEAT. 

For  a  discussion  of  the  error  in  assuming  for  different  bodies 
that  the  radiation  is  similar  to  that  from  a  black  body,  see  the 
above  reference  to  the  Bulletin  of  the  Bureau  of  Standards  and 
Haber,  ''Thermodynamics  of  Gas  Reactions"  pp.  281-291. 

Questions. 

1.  Define  "black  body"  and  show  that  the  electric  furnace  approximately 
satisfied  the  definition. 

2.  Is  any  correction  required  for  the  absorption  of  the  lenses  in  such  an 
optical  pyrometer?     Explain. 


SOUND 


XXX.    THE  VELOCITY  OF  SOUND. 

References— Elementary:  Duff,  §§584,  586,  600,  603-4;  Ames,  pp.  337,  363;  Crew, 
§213;  Edser  (Heat],  pp.  325-9;  Kimball,  §308;  Reed  &  Guthe,  §132; 
Spinney,  §428;  Watson,  §§287-288,  309. — More  Advanced:  Barton,  Text- 
book of  Sound,  §§531-532;  Poynting  &  Thomson  (Sound),  Chap.  VII. 

The  velocity  of  sound  in  a  medium  can  be  found  if  the  wave- 
length, L,  in  the  medium  of  a  note  of  frequency  n  can  be  de- 
termined; for  v  =  n  L.  If  the  me- 
dium be  contained  in  a  tube,  one 
end  of  which  is  closed,  the  closed 
end  must  be  a  node  and  the  open 
end  a  loop.  Hence  the  length  of 
the  tube  must  be  an  odd  number 
of  quarter  wave-lengths.  Such  a 
tube  will  resonate  to  a  fork,  if  the 
wave-length  of  a  natural  vibration 
of  the  pipe  be  the  same  as  the 
wave-length  to  which  the  fork  gives 
rise.  Thus,  if  the  length  of  pipe 
that  resonates  to  a  fork  of  known 
pitch  be  measured,  we  have  the 
means  of  finding  the  velocity  of 
sound. 

A  long  glass  tube  is  mounted  on 
a  stand.  Water  is  introduced  from 
the  bottom,  where  is  attached  a 
rubber  tube  provided  with  a  pinch- 
cock  and  connected  to  a  glass  bottle. 
By  raising  and  lowering  the  bottle, 
water  may  be  brought  to  any  height 
in  the  tube.  The  additional  connec- 
tions represented  in  Fig.  33,  permit  raising  or  lowering  the  water 
level  by  opening  pinch-cocks.  When  A  is  opened  the  tube  fills 

in 


FIG.  33. 


112  SOUND. 

and  the  tube  empties  upon  opening  D.  C  and  B  are  pinch-cocks 
which  regulate  the  rate  of  flow.  The  vessels  should  be  so  large 
compared  with  the  capacity  of  the  tube  that  it  is  not  necessary 
to  transfer  water  frequently  from  the  lower  to  the  upper  vessel. 

If  a  tuning-fork  be  vibrated  above  the  tube,  resonance  will 
first  take  place  when  the  air  column  is  approximately  one- 
quarter  wave-length  of  the  fork,  next  when  three-quarter  wave- 
lengths, etc.  In  reality,  the  first  loop  is  not  exactly  at  the  open 
end  of  the  pipe,  but  a  short  distance  beyond  the  open  end.  The 
distance  between  two  nodes  is  accurately  half  a  wave-length, 
and  it  is  from  this  distance  that  the  wave-length  is  best  deter- 
mined. 

The  tuning-fork  may  be  sounded  by  gently  striking  the  end 
of  the  fork  against  the  knee  or  a  block  of  soft  wood.  The  fork 
should  be  held  above  the  end  of  the  tube,  so  that  the  plane  of  the 
prongs  includes  the  axis  of  the  tube.  Each  node  should  be 
located  very  carefully,  at  least  four  times,  each  location  being- 
tested  both  with  the  water  rising  and  with  the  water  falling,  and 
the  distance  of  each  position  from  the  open  end  noted.  The 
mean  is  taken  as  the  true  distance.  The  whole  should  then  be 
repeated  with  a  fork  of  different  pitch.  Observe  the  tempera- 
ture and  barometric  pressure,  and  measure  the  diameter  of  the 
tube.  With  a  little  practice  one  can  often  locate  nodes  corre- 
sponding to  the  higher  modes  of  vibration  of  the  fork.  This 
should  be  tried  and,  from  their  wave-lengths  and  the  velocity 
of  sound  as  already  determined,  the  pitch  of  these  higher  sounds 
can  be  calculated. 

The  pitch  of  the  forks  used  may  be  determined  by  comparison 
with  a  standard  fork  by  the  method  of  beats.  The  standard  is 
mounted  on  a  resonance  box  and  is  set  in  vibration  by  pulling 
the  prongs  together  with  the  fingers  and  then  releasing  them., 
The  fork  of  unknown  pitch  is  sounded  in  the  usual  way,  and  the 
end  of  the  shank  is  set  upon  the  resonance  box  of  the  standard. 
If  the  two  forks  are  of  nearly  the  same  pitch,  beats  will  be  heard. 
With  a  stop-watch  the  time  of  ten,  fifteen,  or  twenty  beats,  as 
may  be  convenient,  is  several  times  determined.  Dividing  by  the 
time,  we  have  the  number  of  beats  per  second,  and  this  is  the 
difference  of  pitch  of  the  two  forks  To  determine  which  fork 


THE    VELOCITY   OF    SOUND.  113 

is  the  higher,  add  a  little  wax  to  one  prong  of  the  fork  used  in  the 
experiment.  Since  this  increases  the  inertia  of  the  fork,  it  de- 
creases its  pitch.  If  originally  the  two  forks  have  very  nearly 
the  same  pitch,  so  that  there  is  only  a  fraction  of  a  beat  per 
second,  very  small  amounts  of  wax  should  be  added.  A  large 
piece  of  wax  might  change  the  pitch  of  the  fork  from  above  that 
of  the  standard  to  below.  (Time  may  often  be  saved  by  com- 
paring the  fork  with  the  standard  during  the  necessary  delays 
of  the  work  described  below.) 

The  velocity  of  sound  in  carbon  dioxide  may  be  determined 
in  a  similar  manner.  The  water  surface  is  first  lowered  to  the 
bottom  of  the  tube  and  the  tube  is  filled  with  the  gas  from  a 
generator  through  a  small  tube  lowered  just  to  the  water  surface 
(not  below).  The  generator  consists  of  a  tube  filled  with  marble, 
surrounded  by  dilute  hydrochloric  acid.  In  filling  the  generator 
with  marble,  use  only  whole  pieces,  carefully. excluding  any  dust 
or  pieces  small  enough  to  drop  through  into  the  outer  vessel. 
If  the  gas  is  not  evolved  in  sufficient  quantity,  add  hydrochloric 
acid  to  the  outer  vessel.  When  the  air  in  the  tube  has  been 
entirely  displaced  by  the  gas,  a  lighted  match  introduced  into  the 
top  of  the  tube  will  be  extinguished.  The  delivery  tube  is  now 
withdrawn  from  the  resonance  tube,  while  the  gas  still  flows  out 
to  fill  the  volume  occupied  by  the  delivery  tube. 

The  water  surface  is  now  slowly  raised  and  the  nodes  located 
for  one  of  the  forks.  Observations  can  only  be  made  with  the 
water  rising,  for,  when  the  water  surface  is  lowered,  air  enters 
the  tube.  Hence  each  node  can  be  located  but  once.  The  tube 
is  again  filled  and  the  nodes  redetermined  with  the  same  fork. 

From  the  distance  between  the  nodes  and  the  pitch  of  the  fork 
the  velocity  of  sound  is  determined. 

For  each  gas,  find  the  correction  for  the  open  end;  that  is, 
the  displacement  of  the  loop  beyond  the  end  of  the  tube.  Use 
the  mean  position  of  the  highest  node  and  one-fourth  of  the  mean 
wave-length.  Find  what  fraction  this  displacement  is  of  the 
radius  of  the  tube. 

Calculate,  from  the  mean  values  of  the  velocities,  the  velocity 
of  sound  in  each  gas  at  o°  C.  (see  references).  Find  also  the  ratio 
of  the  specific  heats  from  the  velocity  at  o°  C.,  the  standard 


114  SOUND. 

barometric  pressure  (both  in  absolute  units)  and  the  density  as 
given  in  Table  VI. 

Questions. 

1.  Explain  why  (a)  readings  are  made  with  both  rising  and  falling  water, 
(b)  the  plane  of  the  prongs  of  the  fork  must  contain  the  axis  of  the  tube. 

2.  What  is  the  influence  of  atmospheric  moisture  upon  the  velocity  of  sound? 


XXXI.    VELOCITY  OF  SOUND  BY  KUNDT'S  METHOD. 

References  —  Elementary:  Duff,  §607;  Ames,  p.  364;  Crew,  §207;  Kimball,  §335; 
Reed  &  Guthe,  §142;  Watson,  §317.  —  More  Advanced:  Barton,  Text-book  of 
Sound,  §§499-507;  Poynting  &  Thomson  (Sound),  pp.  115-117;  Watson 


A  glass  tube,  A  G,  about  a  meter  long  and  about  3  cm.  internal 
diameter  is  closed  at  one  end  by  a  tight-fitting  piston,  C,  and  at 
the  other  end  by  a  cork  through  which  passes  a  glass  tube  having 
at  one  end  a  loosely  fitting  cardboard  disk,  D  (Fig.  34).  The 
glass  tube  should  be  about  a  meter  long.  A  little  dry  powdered 
cork  is  sprinkled  in  the  tube,  the  stopper  at  G  is  loosened,  and  a 


AC  D  G 

FIG.  34. 

current  of  air,  dried  by  passage  through  several  drying  tubes, 
is  slowly  forced  through  the  hollow  rod  of  the  piston,  C.  The 
stopper  at  G  is  then  replaced  and  the  glass  tube,  F,  is  held  at  the 
center  and  stroked  longitudinally  with  a  damp  cloth.  The 
piston,  C,  is  adjusted  until  the  powder  collects  in  the  sharpest 
attainable  ridges.  These  ridges  will  appear  where  the  pressure 
changes  are  least;  that  is,  at  the  loops.  Measure  carefully  the 
distance  between  the  two  extreme  ridges  and  divide  by  the  num- 
ber of  segments  into  which  the  tube  is  divided.  This  distance 
(between  two*  loops)  f  is  a  half  wave-length  of  the  waves  in 
the  tube.  Disturb  the  powder  and  make  a  new  adjustment  of 
the  piston,  C,  and  a  new  measurement  of  the  half  wave-length. 
Make  a  third  repetition  of  the  adjustments  and  readings. 

Fill  the  tube  with   another  dried   gas,   for  example,   carbon 


VELOCITY   OF    SOUND    BY    KUNDT  S    METHOD.  115 

dioxide,  illuminating  gas,  hydrogen,  oxygen,  or  hydrogen  sulphide, 
and  determine  the  half  wave-length.  If  n  is  the  constant  pitch 
of  the  note  emitted  by  the  glass  tube  and  I  is  the  wave-length  in 
the  gas 

j         Vi      /i 

v  —  nl  .•.  —  =  •=-. 

v2     It 

Since  the  velocity  changes  at  the  same  rate  with  change  of 
temperature  in  all  gases  (see  references),  the  velocity  of  sound  or 
compressional  waves  at  zero  degrees  in  any  other  gas  than  air  can  be 
calculated  from  the  ratio  of  the  wave-lengths  at  a  common  temper- 
ature, and  the  velocity  in  air  at  zero  degrees  (33,200  cm.  per  second) . 
From  the  velocity  of  sound  at  zero  degrees  in  the  gases  other 
than  air  and  the  standard  atmospheric  pressure  calculate  the 
ratio  of  specific  heats,  7  (see  references).  Table  VI  gives  the 
densities  of  the  more  common  gases  and  vapors  at  zero  degrees 
and  a  pressure  of  76  cm.  of  mercury  =  1013200  dynes  per  square 
centimeter. 

Questions. 

1.  Calculate  (a)  the  velocity  of  compressional  waves  in  glass,  (b)  the  elas- 
ticity E.     (Notice  that  each  end  of  the  glass  rod  must  be  a  loop,  and  the  center 
a  node.*  The  density  of  glass  can  be  obtained  from  Table  VIII.) 

2.  Why  must  the  glass  rod  be  set  in  longitudinal  vibration? 

3.  Why  does  the  powder  collect  at  the  loops? 


LIGHT. 


27.  Monochromatic  Light. 

The  simplest  and  most  useful  monochromatic  light  is  the 
sodium  flame.  Sodium  may  be  introduced  into  a  Bunsen  flame 
by  surrounding  the  tube  of  the  burner  with  a  tightly  fitting 
cylinder  of  asbestos  which  has  been  saturated  with  a  strong  solu- 
tion of  common  salt  and  formed  into  cylindrical  shape  by  wrap- 
ping around  the  burner  while  still  damp.  As  the  top  of  the 
cylinder  is  exhausted,  it  should  be  torn  off  and  the  rest  of  the 
tube  pushed  up  into  the  lower  part  of  the  flame.  A  piece  of 
hard-glass  tubing  held  in  the  flame  will  also  give  a  good  sodium 
light. 

Elements  giving  red,  green,  blue,  and  violet  light  will  be  found 
in  Table  XVIII.  Salts  of  these  elements  (e.  g.,  KNO^  SrClz, 
CaClz,  LiCl)  may  be  introduced  into  the  outer  edge  of  a  bunsen 
flame,  or,  preferably,  over  a  Meckle  burner,  either  in  a  spoon 
of  fused  quartz  with  a  central  hole  through  which  part  of  the 
flame  passes,*  in  a  thin  platinum  spoon,  on  copper  gauze,  or  by 
a  piece  of  wood  charcoal  which  has  absorbed  a  solution.  If  a 
very  intense  light  is  not  required,  a  vacuum  tube  is  a  very  satis- 
factory source  (Table  XVIII).  Intense  light  of  one  general 
color  may  be  obtained  by  filtering  sun  light  or  the  light  from  an 
arc  light  through  colored  glassf  or  gelatine.  The  solutions  given 
in  the  accompanying  table  give  much  purer  monochromatic  light. 

LIGHT  FILTERS  (LANDOLT).t 


COLOR 

THICKNESS 
OF  LAYER 

(MM.) 

AQUEOUS  SOLUTION  OF 

GRAMS  PER 
100  C.c. 

AVERAGE 
WAVE-LENGTH 
(ANGSTROM  UNITS) 

Red  

2O 

Crystal  violet  5  BO 

oo^ 

2O 

Potassium  chromate 

10. 

6560 

Green  

20 
20 

Copper  chloride 
Potassium  chromate 

60. 
10. 

5330 

Blue  

20 
20 

Crystal  violet 
Copper  sulphate 

.005 
15- 

4480 

*  Obtainable  from  the  Silica  Syndicate,  London, 
t  Measurement  of  High  Temperatures,  Burgess  &  Le  Chatelier,  p.  135. 
t  Mann,  Manual  of  Advanced  Optics,  p.  185. 

116 


SPHERICAL  MIRRORS   AND   LENSES.  117 

28.  Rule  of  Signs  for  Spherical  Mirrors  and  Lenses. 

Mirrors.  —  Consider  the  side  upon  which  the  incident  light  falls 
as  the  positive  side  of  the  mirror.  If  the  object,  the  image,  or 
the  principal  focus  is  on  this  side,  their  respective  distances, 
(#,  v,  f  =  r/2)  will  be  positive;  if  on  the  other  side,  negative. 
Therefore,  the  focal  length  (and  hence  radius)  is  positive  for 
concave  mirrors  and  negative  for  convex.  The  object  distance, 
u,  will  obviously,  in  most  cases,  be  positive. 

The  formula  for  all  spherical  mirrors  is: 


11     v     f     r 

if  the  signs  of  the  numerical  quantities  which  are  substituted 
for  Uj  v,f,  and  r  are  determined  by  the  above  rule. 

Lenses.  —  Let  all  the  distances,  «,  i>,  /,  ri,  r2  be  positive  for  the 
double  convex  lens,  when  the  object  is  outside  the  principal  focus; 
that  is,  in  the  most  common  case.  The  formula  for  all  lenses  is 
then 


i.i  I  ,  , /I  .  i\ 
— h- =  -7=(w-i)l  -H —  I 
u  v  f  \r.l  r$J 


As  an  illustration  of  the  application  of  this  rule,  consider  the 
signs  of  these  distances  when  an  image  of  a  real  object  is  formed 
by  a  double  concave  lens.  The  distance,  u,  of  the  object  is 
obviously  measured  on  the  same  side  as  it  would  be  in  the  stand- 
ard case  of  the  double  convex  lens  and  is,  there fo re r  positive. 
The  distances  /  and  v  are,  however,  measured  on  the  same  side 
of  the  lens  as  the  object,  or  opposite  to  the  standard  case  with  the 
double  convex  lens,  and  are,  therefore,  negative,  ri,  the  radius 
of  the  front  face,  is  on  the  same  side  as  the  object,  while  in  the 
case  of  the  double  convex  lens  this  radius  is  on  the  other  side, 
therefore  TV and  similarly  r2,  is  negative  for  a  double  concave  lens. 


Il8  LIGHT. 


XXXII.  PHOTOMETRY. 

References — Elementary:  Duff,  §§628,  712;  Ames,  pp.  437,  442;  Edser  (Light} 
pp.  9-20;  Kimball,  §§796-800;  Spinney,  Chap.  XLV;  Watson,  §§361-364.— 
More  Advanced:  Palaz's  Photometry;  Stine's  Photometrical  Measurements; 
Watson  (Pr.),  §§154-156. 

The  intensity  of  illumination  of  a  surface  by  a  source  of  light 
of  small  area  varies  inversely  as  the  square  of  the  distance. 
Hence  it  follows  that,  if  two  lights  produce  equal  intensities  of 
illumination  at  a  point,  P,  their  illuminating  powers,  or  the 
intensities  of  illumination  they  can  produce  at  unit  distances, 
are  directly  as  the  squares  of  their  distances  from  P.  This  is 
the  basis  of  all  practical  methods  of  comparing  illuminating 
powers. 

As  a  means  of  testing  when  two  different  sources  of  light  pro- 
duce equal  illumination  at  a  point,  various  so-called  screens  have 
been  used.  The  one  that  has  been  most  extensively  employed 
is  Bunsen's  grease-spot  screen.  It  is  based  on  the  fact  that  a 
grease-spot  on  paper  is  invisible  when  the  paper  is  equally 
illuminated  on  both  sides,  since  viewed  from  one  side  as  much 
light  is  gained  by  transmission  from  the  farther  side  as  is  lost 
by  transmission  to  the  farther  side. 

Another  screen  more  perfect  in  some  respects  is  that  of  Lummer 
and  Brodhun.  A  white  opaque  disk  (see  Fig.  35)  is  illuminated 
on  opposite  sides  by  the  two  sources  of  light.  An  arrangement 
of  mirrors  and  lenses  enables  one  eye  to  view  both  sides  at  once. 
Two  plane  mirrors  reflect  rays  from  the  two  sides  into  a  double 
glass  prism.  This  consists  of  two  separate  right-angled  prisms, 
the  largest  face  of  one  being  partly  beveled  away  and  the  two 
being  cemented  together  by  Canada  balsam,  which  has  the  same 
optical  density  as  the  glass,  and  therefore  reflects  no  light.  The 
central  rays  from  the  left  pass  through  the  double  prism  to  the 
telescope  while  the  marginal  rays  are  totally  reflected  by  the 
beveled  edge.  The  marginal  rays  from  the  right  are  totally 
reflected  and  reach  the  telescope,  but  the  central  rays  pass 
through.  Thus  the  eye  sees  a  circular  portion  of  the  left  side 
of  the  opaque  disk  and  a  surrounding  rim  of  the  right  side. 

To  eliminate  error  from  lack  of  symmetry,  the  lamps  compared 


PHOTOMETRY. 

should  be  interchanged  in  the  course  of  the  readings  or  the  screen 
should  be  rotated  180°. 

The  lights  to  be  compared  are  mounted  at  opposite  ends  of  a 
graduated  bar  3  meters  long,  which,  with  a  parallel  bar  and  suit- 
able supports,  constitutes  the  photometer  bench.  The  screen 
is  mounted  on  a  carriage  movable  along  the  bench. 

Many  light  standards  have  been  employed.  A  candle  of 
certain  carefully  specified  dimensions  was  long  employed,  and  the 
illuminating  power  of  such  a  candle  is  still  regarded  as  the  unit 
and  called  "one  candle-power,"  but,  for  practical  purposes  in 
testing,  some  other  standard  is  usually  employed.  The  best 
such  standard  is  a  lamp,  with  a  wick  of  specified  form  and  di- 
mensions, burning  amyl  acetate  with  a  flame  of  specified  height. 

(See  references.)      Its  relation  to  the  _          

"  candle-power "  is  I  c.  p.  =  1.14  amyl 
acetate  units.  For  most  purposes  an 
incandescent  lamp  that  has  been  stand- 
ardized is  the  most  useful  standard, 
especially  in  the  study  of  incandescent 
lamps;  but  it  must  not  be  used  any 
great  length  of  time  without  being 
re-standardized  since  its  illuminating 
power  changes  with  prolonged  use. 

The  chief  difficulty  in  comparing  two 

different  forms  of  light  is  due  to  the  fact  that  a  difference  of  quality 
of  the  two  lights  renders  perfectly  equal  and  similar  illumination  of 
the  two  sides  of  the  screen  impossible.  This  difficulty  is  still  more 
marked  in  the  study  of  arc-lights  (for  mechanical  arrangements  see 
Stine,  p.  236),  for  which  it  is  best  to  use  as  an  intermediate  unit  a 
very  powerful  incandescent  lamp.  The  latter  may  be  standardized 
by  comparison  with  an  ordinary  incandescent  lamp,  which  again 
is  compared  with  an  amyl  acetate  standard.  Before  connecting 
a  lamp  to  a  circuit,  ascertain  that  the  voltage  is  not  excessive  for  that 
particular  lamp. 

(A)  Carefully  standardize  an  incandescent  lamp,  for  use  as  a 
working  standard,  by  comparison  with  either  an  amyl  acetate 
lamp  or  a  standardized  incandescent  lamp.  If  the  latter  is  used, 
the  lamps  should  be  in  parallel,  that  the  voltage  may  be  the  same, 


I2O  LIGHT. 

and  a  variable  resistance  should  also  be  in  the  circuit,  by  varying 
which  the  voltage  across  the  lamps  is  maintained  at  the  value 
prescribed  for  the  standard.  See  that  the  filament  of  the  stand- 
ard is  in  the  marked  azimuth  and  note  the  position  of  the  filament 
for  which  the  other  lamp  is  standardized. 

In  each  case  several  settings  of  the  screen  should  be  rapidly 
made,  and  then  the  screen  reversed  and  several  more  made. 
The  calculations  may  be  facilitated  by  Table  XXI. 

(B)  The  law  of  inverse  squares  should  be  tested  by  comparing 
two  somewhat  different  incandescent  lamps  (i)  when  3  m.  apart, 
(2)  when  2.5  m.  apart,  (3)  when  2  m.  apart  on  the  photometer 
bench.     The  ratio  of  their  illuminating  powers,  as  deduced  in  the 
three  cases,  should  be  a  constant. 

(C)  The  horizontal  distribution  of  candle-power  about  an  in- 
candescent lamp  should  be  studied.     This  incandescent  lamp 
should  be  connected  in  parallel  with  the  working  standard  and 
the  voltage  maintained  at  the  value  for  which  the  latter  was 
standardized.     The  lamp  should  be  mounted  on  the  revolving 
lamp-holder  of  the  photometer,  care  being  taken  to  have  the 
center  of  the  filament  at  the  same  height  as  the  center  of  the 
screen.     The  lamp  is  first  turned  to  the  standard  position,  i,  e., 
the  position  in  which  the  plane  of  the  shanks  of  the  filament  is 
at  right  angles  to  the  photometer  bench,  and  a  marked  face  of 
the  lamp  is  toward  the  screen.     The  candle-power  of  the  lamp 
is  to  be  found  in  this  position  and  at  posftions  30°  apart  as  the 
lamp  is  rotated  through  360°.     Two  careful  readings  should  be 
made  at  each  angle  and  the  c.  p.  deduced  from  the  mean.     The 
mean  of  all  these  values  of  the  c.  p.  gives  the  mean  horizontal 
candle-power.     A  curve  should  be  plotted,  giving  the  distribution 
of  c.  p.  in  polar  co-ordinates.     The  mean  horizontal  candle-power 
is  more  easily  determined  by  continuously  rotating  the  lamp 
about  a  vertical  axis  by  means  of  a  small  motor. 

(D)  Efficiency  of  an  Incandescent  Lamp. — Keeping   the   po- 
tential of  the  working  standard  at  the  proper  point  (or  calibrating 
and  using  a  lamp  which  may  be  connected  to  the  lighting  circuit 
if  the  potential  of  that  is  constant),  apply  various  potentials  to 
the  lamp  used  in  (C)  at  intervals  between  about  25%  below  the 
normal  voltage  to  25%  above.     For  each  potential,  determine 


PHOTOMETRY.  121 

the  candle-power  and  current.     Calculate  the  watts  consumed 
and  the  watts  per  candle-power. 

In  the  report,  plot  in  three  curves,  with  volts  as  ab^issae, 
(a)  current,  (b)  candle-power,  (c)  watts  per  candle-power. 
The  scales  of  the  three  curves  should  be  shown  on  the  vertical 
axis. 

(£)  Mean  Spherical  Candle-power. — With  the  lamp  in  the  standard  posi- 
tion of  (C),  find  the  c.  p.  at  intervals  of  30°  in  a  vertical  circle  by  rotating  the 
lamp  about  a  horizontal  axis.  After  this,  start  again  from  the  standard  posi- 
tion and  first  turn  the  lamp  through  45°  in  azimuth  (or  around  a  vertical  axis), 
and  then,  as  before,  find  the  c.  p.  at  intervals  of  30°  in  the  vertical  circle  of  45° 
azimuth,  and  so  for  the  vertical  circles  of  90°  and  135°  azimuth.  As  before, 
plot  the  curves  of  distribution  in  polar  co-ordinates. 

To  find  the  mean  spherical  candle-power  omit  any  repetitions  and  take  the 
mean  of  the  readings  in  the  following  positions: 

1.  At  tip i 

2.  At  60°,  120°,  240°,  300°  on  the  vertical  circles  of  o°  and  90°  azimuth ...       8 

3.  At  30&,  150°,  210°,  330°  on  the  vertical  circles  of  o°,  45°,  90°,  135°  azi- 

muth        16 

4.  12  equidistant  positions  on  horizontal  circle 12 

5.  At  base  (o) i 

Total 38 

These  directions  are  chosen  because  they  are  nearly  uniformly  distributed 
in  space. 

(F)  If  time  permit,  study  the  differences  of  quality  of  light 
given  by  different  sources;  e.  g.,  compare  an  oil  lamp  and  an 
incandescent  lamp  using  interposed  colored  glasses;  (i)  a  pair 
of  red  glasses,  (2)  of  yellow  glasses,  (3)  of  blue  glasses.  Calculate 
the  relative  illuminating  powers  in  each  case. 

The  possible  error  may  be  deduced  as  usual  from  the  mean 
deviation  of  the  readings  in  a  set. 

Questions. 

1.  Prove  the  inverse  square  law. 

2.  Explain  the  deviation  of  the  current-voltage  curve  from  a  straight  line. 

3.  What  is  the  advantage  in  increasing  the  voltage  applied  to  an  incan- 
descent lamp?     Disadvantage? 


122  LIGHT. 

XXXIII.    SPECTROMETER  MEASUREMENTS. 

References— Elementary:  Duff,  §§656,  659-664,  709;  Ames,  pp.  459-460,  505- 
506;  Crew,  §§500-502;  Edser  (Light),  pp.  86-91;  Kimball,  §§848,  896;  Reed 
&  Guthe,  §§450,  479;  Spinney,  §§495~496;  Watson,  §§341,  357-— More 
Advanced:  Baly's  Spectroscopy,  Chap.  VI;  Mann's  Manual  of  Advanced 
Optics,  Chap.  VII;  Wood's  Physical  Optics,  pp.  74-75. 

A  spectrometer  consists  of  a  framework  supporting  a  telescope 
and  a  collimator,  both  movable  about  a  vertical  axis,  and  a  plat- 
form movable  about  the  same  axis.  The  platform  is  for  support- 
ing a  prism  or  grating.  The  collimator  is  a  tube  containing  an 
adjustable  slit  at  one  end  and  a  lens  at  the  other  end.  The  pur- 
pose of  the  collimator  is  to  render  light  coming  from  the  slit 
parallel  after  it  leaves  the  lens.  (Only  when  the  light  that  falls 
on  a  prism  is  parallel  light,  that  is,  light  with  plane  wave  front, 
does  it  seem  when  emerging  from  the  prism  to  come  from  a  clearly 
defined  source.  When  it  is  not  parallel,  there  is  spherical 
aberration.)  Hence  the  slit  of  the  collimator  should  be  in  the 
principal  focus  of  the  lens.  The  telescope  is  for  the  purpose  of 
viewing  the  light  that  comes  from  the  collimator,  either  directly 
or  after  the  light  has  been  refracted  or  reflected.  Hence,  since 
the  light  that  comes  from  the  collimator  is  supposed  to  be  parallel, 
that  is,  as  if  it  came  from  a  very  distant  source,  it  follows  that  if 
the  telescope  is  to  receive  the  light  and  form  a  distinct  image  of 
the  slit,  the  telescope  must  be  focused  as  for  a  very  distant  object 
(theoretically  an  infinitely  distant  one.) 

The  first  adjustment  is  to  focus  the  telescope.  First  focus  the 
eye-piece  of  the  telescope  on  the  cross-hairs  and  then  focus  the 
whole  telescope  on  a  distant  object  out  of  doors.  The  telescope 
will  now  be  in  focus  for  parallel  rays.  Turn  the  telescope  to 
view  the  image  of  the  slit  formed  by  the  collimator  and  adjust 
the  slit  until  its  image  is  seen  most  distinctly. 

That  the  instrument  should  be  in  complete  adjustment,  it  is 
necessary  that  the  telescope,  collimator,  and  platform  should 
rotate  about  the  same  axis,  and  that  the  optical  axes  of  the  tele- 
scope and  the  collimator  should  be  perpendicular  to  this  axis  of 
rotation.  For  fine  work  spectrometers  are  made  with  all  these 
parts  separately  adjustable,  but  simpler  instruments  have  the 


SPECTROMETER   MEASUREMENTS.  123 

telescope  and  collimator  put  into  permanent  adjustment  by  the 
instrument  maker.  In  any  case  the  telescope  and  collimator 
should  not  be  adjusted  for  level  without  the  advice  of  an  in- 
structor. 

Adjustment  of  Prism. — The  refracting  edge  of  the  prism  must 
be  made  parallel  to  the  axis  of  the  instrument.  Turn  the  slit 
horizontal  and  adjust  its  width  until  one  edge  of  the  image  of 
the  slit  falls  on  the  horizontal  cross-hair  when  telescope  and  col- 
limator are  in  line.  Place  the  prism  on  the  platform  with  one  of 
the  faces  perpendicular  to  the  line  joining  two  of  the  leveling 
screws.  Place  the  telescope  so  as  to  receive  the  image  of  the  slit 
reflected  from  this  face  of  the  prism.  Adjust  these  two  leveling 
screws  until  the  image  of  the  proper  edge  of  the  slit  coincides  with 
the  horizontal  cross-hair.  Then  observe  the  image  reflected 
from  the  other  face  and. adjust  the  third  leveling  screw  until  the 
edge  of  this  image  is  on  the  horizontal  cross-hair.  A  little  con- 
sideration will  show  that,  when  these  two  adjustments  have 
been  made,  both  faces,  and  therefore  the  refracting  edge,  are 
parallel  to  the  axis.  Restore  the  collimator  slit  to  the  vertical 
position. 

Measurement  of  the  Angle  of  a  Prism. — Method  (A). — The 
prism  should  be  so  placed  that  the  faces  are  about  equally  inclined 
to  the  collimator.  To  secure  good  illumination,  the  edge  of  the 
prism  should  be  near  the  axis  of  the  instrument.  The  telescope 
is  turned  to  view  the  image  of  the  slit  in  the  two  faces  alternately, 
and  the  scale  and  vernier  read  when  the  slit  and  cross-hair 
coincide,  the  slit  being  narrowed  until  barely  visible.  If  the 
scale  is  provided  with  two  verniers,  to  eliminate  error  from 
eccentricity,  always  read  them  both.  Half  of  the  angle  between 
the  two  positions  of  the  telescope  gives  the  angle,  A,  of  the  prism, 
as  may  be  readily  seen  by  drawing  a  diagram.  The  readings  on 
each  side  should  be  repeated  three  times. 

Method  (B). — The  following  method,  which  is  sometimes* 
easier  than  the  preceding,  may  be  used  if  the  platform  that  carries 
the  prism  can  be  rotated  and  the  rotation  read  by  a  scale.  Turn 
one  face  of  the  prism  so  as  to  reflect  the  image  of  the  slit  into  the 
telescope.  Adjust  the  telescope  until  the  vertical  cross-hair 
coincides  with  the  slit  and  then  read  the  platform  scale.  Now 


124  LIGHT. 

rotate  the  platform  until  the  other  face  of  the  prism  reflects  the 
slit  and  again  read  the  platform  scale.  The  difference  of  the 
readings  is  180  =±=  A,  as  may  readily  be  seen  by  drawing  a  figure. 
The  observations  should  be  repeated  at  least  three  times. 

We  are  now  in  a  position  to  make  a  final  measurement  for  find- 
ing the  index  of  refraction  of  the  glass  of  the  prism  for  any  par- 
ticular light  of  the  spectrum,  for  instance,  sodium  light  (see  p. 
1  1  6).  The  only  additional  measurement  necessary  is  the  devia- 
tion produced  by  the  prism  when  it  is  in  such  a  position  that 

it  gives  a  minimum  deviation 
to  the  light  refracted  through 
it. 

Minimum  Deviation.  —  The 
position  of  minimum  deviation 
is  such  that  the  image  of  the  slit 
seen  in  the  telescope  moves  in 

the  same  direction  (that  of  increasing  deviation)  no  matter  which 
way  the  platform  carrying  the  prism  is  turned.  There  are,  of 
course,  two  positions  in  which  the  deviation  can  be  obtained,  one 
with  the  refracting  edge  turned  toward  the  right  of  the  observer, 
and  the  other  with  it  toward  the  left.  The  deviation  in  each  case 
is  the  angle  between  the  corresponding  position  of  the  telescope 
and  its  position  when  looking  directly  into  the  collimator,  the 
prism  being  removed.  But  it  is  not  necessary  to  remove  the 
prism,  for  it  is  easily  seen  that  the  minimum  deviation  must  also 
be  equal  to  half  of  the  angle  between  the  two  positions  of  the 
telescope  when  observing  the  minimum  deviation.  These  two 
positions  should  be  observed  three  times  successively,  and  the 
mean  value  for  the  minimum  deviation,  D,  taken.  From  A  and 
D  the  index  of  refraction  may  be  calculated  by  the  formula 


If  time  permit,  determine  the  index  of  refraction  for  as  many 
other  wave-lengths  (colors)  as  possible  (see  p.  116). 

The  possible  error  of  the  determination  of  the  refractive  index 
can  be  calculated  by  means  of  formulae  deduced  by  the  calculus, 
as  explained  on  pp.  6-9.  A  simple,  but  less  accurate  method  is 


MEASUREMENTS  OF  RADIUS  OF  CURVATURE.         125 

to  recalculate  n  with  A  and  D,  increased  by  their  mean  devia- 
tions and  to  consider  the  difference  between  this  value  and  the 
original  value  as  the  possible  error. 

It  is  probable  that  in  this  experiment  there  are  other  sources 
of  error  that  exceed  mere  error  in  reading  the  scale;  e.  g.,  (i) 
The  faces  of  the  prism  may  not  be  true  planes,  (2)  the  divided 
circle  may  not  be  uniform,  (3)  the  center  of  the  circular  scale  may 
not  coincide  with  the  center  of  the  instrument,  (4)  the  various 
adjustments  may  not  be  perfect,  (5)  there  may  be  difficulty  in 
fixing  the  position  of  minimum  deviation.  These  errors  might  be 
eliminated  by  repeating  all  the  adjustments  and  observations 
many  times  and  using  different  parts  of  the  divided  scale.  There 
is  no  other  way  of  allowing  for  them. 

Questions. 

1.  Give  both  physical  and  mathematical  definitions  of  the  refractive  index. 

2.  Why  is  monochromatic  light  used? 

3.  Why  is  the  minimum  deviation  chosen? 

4.  \Vhat  is  the  approximate  relation  between  the  index  of  refraction  and 
the  wave  length? 

5.  Is  the  dispersion  greater  for  long  or  short  waves?     Explain. 


XXXIV.    MEASUREMENT  OF  RADIUS  OF  CURVATURE. 

References — Edser  (Light],  pp.  1 16-121 ;  Glazebrook  and  Shaw,  Practical  Physics, 
pp.  339-343;  Kohlrausch's  Physical  Measurements,  pp.  174-176. 

The  radius  of  curvature  of  a  surface  may  be  determined  from 
the  size  or  position  of  the  image  which  the  spherical  surface, 
regarded  as  a  mirror,  forms  of  a  definite  object.  Method  (A)  be- 
low is  especially  applicable  to  the  measurement  of  the  radius  of 
curvature  of  convex  surfaces,  and  method  (B)  to  concave  surfaces. 

(A)  Two  bright  objects  (see  Fig.  37)  are  placed  on  a  line  at 
right  angles  to  the  axis  of  the  spherical  surface,  the  intersection 
of  the  line  and  the  axis  being  at  a  considerable  distance  A ,  from 
the  surface,  and  each  object  being  at  a  distance  L/2  from  the 
axis.  If  the  apparent  distance  between  the  images  of  the  two 
objects  be  /,  the  radius  of  curvature  of  the  surface  is 

2AI 


126  LIGHT. 

the  +  sign  being  used  for  a  concave  surface  and  the    —  sign  for 
a  convex. 

Proof. 

(For  convex  mirror.) 

Let  d  =true  distance  between  the  images,  x  =  distance  of  images  from  mirror. 
By  geometry 

L  _ A±r      d  _ A+x         L _ (A+r)(A+x) 
d=7~-x'     T=    A    '      "T*      (r-*M      ' 
By  the  equation  for  spherical  mirrors  (see  p.  117). 

(i)  1_1,2 

x~A  +  r 

-  -  'i-i-ii 

•  x     r~A  +  r 
r—x    A-\-r 
rx   ~  Ar 
L_A+x_        A 
•'•   /-"     x     ~1+~x- 
From  (i)  A  2 A 

*  =  I+T 

L  2A 

•;  T-'+T* 

2AI 


The  radius  of  curvature  should  be  found  for  both  surfaces  of  a 
double  convex  lens.  The  lens,  perferably  the  one  used  in  Exp. 
XXXV,  if  that  has  already  been  performed,  is  fitted  in  a  clamp 
in  a  darkened  recess.  At  some  distance  are  two  vertical  slits 

illuminated  from  behind  by  in- 
candescent lamps  (or  the  lamp 
filaments  themselves  may  be 
'x--^j>o  used)  and  between  them  a  tele- 
scope. The  telescope  and  the 
lens  are  adjusted  until,  on  looking 


FlG    7  through  the  telescope  toward  the 

lens,  the  illuminated  slits  are  seen 

reflected  from  the  near  surface  of  the  lens.  Distinguish  these  im- 
ages from  the  images  produced  by  the  rear  surface  of  the  lens  by 
the  change  of  focus  necessary  to  make  one  pair  of  images  most 
distinct,  and  then  to  make  the  other  pair  most  distinct,  or,  by 
observing  the  two  images  of  a  light  held  just  outside  one  of  the 


MEASUREMENT  OF  RADIUS  OF  CURVATURE.        127 

slits.  Remember  that  the  telescope  inverts.  A  scale  is  mounted 
over  the  lens  so  that  the  upper  edge  is  just  below  the  center 
of  the  lens.  The  telescope  is  focused  upon  the  scale,  and  rotated 
until  the  vertical  cross-hair  bisects  one  of  the  images  from  the 
near  surface  of  the  lens,  and  the  scale  read  where  crossed  by  the 
cross-hair.  (Estimate  tenths  of  millimeters  as  always.)  A  simi- 
lar reading  is  made  for  the  other  image.  The  difference  between 
the  two  readings  gives  the  apparent  distance,  /,  between  the  im- 
ages. At  least  six  independent  determinations  of  this  distance 
should  be  made.  Measure  the  distance,  L,  between  the  slits,  the 
distance,  A,  from  the  lens  surface  to  the  line  joining  the  slits  and 
substitute  in  the  formula. 

From  the  two  radii  of  curvature  and  the  focal  length,  if  known, 
calculate  the  refractive  index,  n,  of  the  glass  of  the  lens  by  means 
of  the  formula 


M, 


(seep.  117). 


(B)  As  a  concave  mirror  we  may  use  one  of  the  surfaces  of  a 
concave  lens,  mounted  in  a  lens-holder.  To  reduce  reflection 
from  the  other  surface,  the  latter  may  be  covered  by  moist  filter 
paper.  The  radius  is  determined  from  u,  the  distance  of  the  ob- 
ject, v,  the  distance  of  the  image  and  the  formula 

I.I       2     , 

-+-=-,  (see  p.  117). 

In  locating  the  image,  use  is  made  of  the  fact  that  if  the  eye 
is  a  considerable  distance  off,  a  real  image  can  be  seen  in  space 
as  well  as  a  virtual  image,  and  a  wire,  needle,  or  pointer  is  moved 
about  until  there  is  no  parallax  between  it  and  the  image;  i.  e., 
until,  when  the  eye  is  moved  about,  there  is  no  relative  motion 
of  the  two. 

A  vertical  wire  illuminated  by  a  lamp,  behind  which  is  a  sheet 
of  white  paper,  is  a  convenient  object,  and  a  second  mounted 
wire  is  moved  about  until  it  coincides  with  the  image  of  the  first 
(see  (B)  Exp.  XXXV)  .  The  image  should  be  found  for  at  least  the 
following  three  typical  positions  of  the  object.  For  each  position 
make  several  settings  and  from  the  means  determine  u  and  v, 
and  from  them  determine  r. 


128  LIGHT. 

(1)  Let  the  object  be  at  a  considerable  distance  from  the 
mirror. 

(2)  Let  the  object  be  at  the  center  of  curvature  of  the  mirror. 
In  this  position  the  image  and  the  object  coincide. 

(3)  Let  the  object  be  within  the  principle  focus.     For  this 

position  the  wire  locating  the  image  must  be  on  the 
other  side  of  the  lens.  This  wire  is  moved  about 
until  the  prolongation  above  the  lens  of  the  image 
of  the  first  wire  coincides  with  what  is  seen  of  the 
second  wire  above  the  lens. 

In  the  report,  sketch  the  relative  positions  of 
mirror,  image,  and  object,  and  state  whether  the 
image  was  magnified  of  diminished,  erect  or  in- 
verted. 

FIG  '38  (Q   ^  time  permit,  check  your  results  with  a 

spherometer  (see  p.  15).  The  spherometer  should 
be  read  alternately  on  a  plane  surface  and  on  the  lens.  Let 
a  =  difference  in  the  two  readings  (see  Fig.  38)  and  r  =  radius  of 
the  circle  of  the  legs.  Then  the  radius  of  curvature  of  the  lens  is 


2a 
as  may  be  easily  shown. 

Questions. 

1.  For  what  lenses  would  the  first  method  of  determining  the  radius  of 
curvature  be  preferable,  and  when  would  the  spherometer  be  preferable? 

2.  What  objection  is  -there  to  determining  the  radius  of  curvature  of  the 
farther  face  of  a  convex  lens,  considering  it  a  concave  surface? 

3.  What  advantages  has  the  method  used  in  (B)  for  locating  real  images 
over  the  use  of  a  screen? 

4.  How  could  you  directly  determine  with  a  screen  the  center  of  curvature 
of  a  concave  mirror? 


XXXV.    FOCAL  LENGTH  OF  A  LENS. 

References — Elementary:  Duff,  §§666-671,  676;  Ames,  pp.  470-483;  Crew,  §§459- 
462,  464;  Edser  (Light),  pp.  110-116;  Kimball,  §§850,  866;  Reed  &  Guthe, 
§§453-469;  Spinney,  §§479-483,  49O-494;  Watson,  §§34^-349-  $71— More 
Advanced:  Glasebrook  &  Shaw,  Practical  Physics,  pp.  343-352. 

The  focal  length  of  a  lens  is  the  distance  from  the  optical  cen- 
ter of  the  lens  to  the  focus  for  rays  of  light  from  an  infinite  dis- 


FOCAL  LENGTH   OF  A   LENS.  129 

tance;  i,  e.,  for  plane  waves.  If/  is  the  focal  length,  u  the  dis- 
tance of  the  object  from  the  lens,  and  v  that  of  the  image,  then, 
with  the  convention  respecting  signs  given  on  page  1  1  7,  for  all  lenses 


= 

v     u     f 

(A)  Real  Image.  —  (If  Exp.  XXXIV  has  preceded,  use  the  same 
lenses.)     An  "object,"  the  lens,  and  a  screen   for  receiving  the 
image  of  the  object,  are  mounted  so  that  they  can  be  moved  along 
a  graduated  scale.    A  convenient  form  for  the  object  is  a  wire  cross 
or  gauze,  mounted  in  a  black  wooden  support,  and  illuminated 
from  behind  by  an  incandescent  lamp.    The  lens  is  clamped  in  a 
frame  movable  along  the  scale.     This  should  grasp  the  lens  on 
the  sides,  leaving  the  top  and  bottom  clear.    The  distance  from 
the  center  of  the  lens  to  the  point  used  as  an  index  in  locating  the 
position  of  the  support  on  the  scale  must  be  applied  as  a  correc- 
tion to  the  readings.    With  object  and  screen 
in  fixed  positions  that  are  recorded,  the  lens  is 
adjusted   until  the  image  on  the  screen  is  as 
distinct  as  possible  and  its  position  is  then  re- 
corded.    This  should   be  done   several   times, 
and  the  mean  taken  for  the  position  of  the  lens. 
Keeping  object  and  screen  fixed  and  moving  pia  39> 

the  lens  about,  another  image  will  be  found,  for 
which  similar  observations  should  be  made.     Calculate  /  from 
the  averages  of  all  the  values  of  u  and  v.    The  object  and  screen 
should  then  be  shifted  and  the  observations  repeated.    From  the 
two  sets  of  observations  a  mean  value  of/  is  deduced. 

Study  of  Spherical  Aberration.  —  Determine  /  for  the  central 
part  of  the  lens  by  covering,  with  a  pasteboard  screen,  all  but 
a  central  disk  of  about  one-third  the  diameter  of  the  lens.  Sim- 
ilarly determine  /  for  the  edge  of  the  lens,  using  a  diaphragm 
covering  all  but  the  edge. 

Study  of  Chromatic  Aberration.  —  Using  the  entire  lens,  deter- 
mine/for red  light  by  placing  red  glass  before  the  lens  or  object, 
and  similarly  for  blue  or  green  light. 

In  the   report,  tabulate,  for  comparison,  the  different  mean 
values  of  /. 
9 


130  LIGHT. 

(B)  Virtual  Image. — In  the  preceding  a  real  image  was  ob- 
served, but  the  focal  length  may  also  be  found  from  observations 
of  a  virtual  image.  The  following  directions  apply  to  a  diver- 
gent lens.  A  vertical  dark  line  on  a  white  background  serves  as 
object.  The  image  (between  the  lens  and  the  object)  is  located 
with  a  short  vertical  wire,  which  is  moved  back  and  forth  until 
a  position  is  found  where  the  image  of  the  dark  line  seen  through 
the  lens  (a  in  Fig.  39)  appears  at  the  same  distance  as  the  portion 
of  the  wire  seen  just  below  the  lens  (b  in  Fig.  39).  This  is  secured 
when  there  is  no  relative  motion  of  the  image  and  this  wire  as  the 
eye  is  moved  horizontally,  i.  e.,  the  wire  appears  as  the  prolonga- 
tion of  the  image  of  the  dark  line  or  remains  equidistant  from 
such  a  prolongation,  v  will  be  the  distance  from  the  center  of 
the  lens  to  this  wire  which  locates  the  image.  Using  a  longer  wire 
as  the  object  and  the  dark  line  to  locate  the  image,  this  method 
may  be  applied  to  the  virtual  image  of  a  convergent  lens. 

Estimate  the  possible  error  of  a  typical  measurement  of  /. 
Since  practically  all  the  error  is  in  the  location  of  the  lens,  the 
distance  between  the  object  and  the  screen  may  be  considered 
free  from  error.  If  this  distance  is  designated  by  w,  the  formula 
becomes 


u     w-u    /' 

from  which  a  formula  may  easily  be  derived  for  the  possible 
error  in  /  in  terms  of  the  possible  error  in  u.  The  latter  may  be 
taken  as  the  mean  deviation  from  the  mean  in  the  location  of  the 
lens  (see  p.  4). 

If  Exp.  XXXIV  has  preceded,  determine  the  refractive  index 
of  the  glass  from  the  focal  length  and  the  radii  of  curvature. 


Questions. 

1.  What  is  the  minimum  distance  between  object  and  screen  to  secure  a 
real  image?     The  maximum  distance  between  object  and  lens  to  secure  a 
virtual  image? 

2.  What  advantage  is  there  in  covering  with  a  diaphragm  all  but  the  central 
portion  of  a  lens?     What  disadvantage? 

3.  What  is  the  cause  of  chromatic  aberration? 

4.  What  sort  of  a  lens  would  show  large  spherical  aberration?     Large 
chromatic  aberration? 


LENS   COMBINATIONS.  13! 

XXXVI.   LENS  COMBINATIONS. 

References—  D  rude'  s  Optics,  pp.  44-46,  66-72;  Edser  (Light),  Chaps.  VI,  VII, 
X;  Hastings  Light,  Appendix  A;  Watson  (Pr),  pp.  358-367- 

(A)  Determination  of  Principal  Foci.  Calculation  from  Focal 
Lengths  and  Separation.  —  Let  two  lenses  of  focal  lengths,  /i,  and 
/2,  be  separated  a  distance  d.  An  object  at  a  distance  u  from  the 
first  lens  forms  an  image  at  a  distance  v  determined  by  the  equa- 
tion 


V      fi      II         Ufi 


This  image  acts  an  as  object  for  the  second  lens  at  a  distance 
d  —  v.  Hence  the  distance  of  the  final  image  from  the  optical 
center  of  the  second  lens  is  given  by  the  question 


I        ! 


v'    /2     d—  v    /2     du  —  dfi  —  ufi 

If  u  is  infinite,  v'  is  the  distance,  F,  of  the  principal  focus  from 
the  optical  center  of  the  second  lens, 


Experimental  Location.  —  Determine  experimentally  the  posi- 
tion of  this  principal  focus  by  finding  the  position  in  which  an  ob- 
ject must  be  placed  for  clear  vision  when  it  is  viewed  through  the 
combination  with  a  telescope  focused  for  a  very  distant  object. 
Compare  with  the  calculated  position.  Determine  similarly  the 
other  principal  focus. 

(B)  Determination  of  Focal  Length.  —  To  determine  the  focal 
length,  the  position  of  the  "principal  points"  must  be  known,  as 
well  as  the  principal  foci.  With  thin  lenses,  the  principal  points 
practically  coincide  at  the  so-called  "optical  center."  In  thick 
lenses  or  lens  combinations  they  may  be  considerably  separated. 
The  focal  length  is  the  distance  from  either  principal  focus  to  the 
nearer  principal  point.  The  equations  for  locating  the  image, 

1+1=1 

' 


132  LIGHT. 

and  for  finding  the  linear  magnification, 


are  applicable  in  all  cases,  if  u  and  v  are  measured  from  the  prin- 
cipal points. 

Since  it  is  difficult  to  locate  the  principal  points,  a  method  is 
often  employed  for  determining  the  focal  length  which  eliminates 
their  position.  Suppose  that  the  linear  magnification  is  Mi,  when 
the  distance  of  the  object  from  the  principal  plane  is  HI,  and  M2 
when  the  object  is  moved  until  its  distance  is  HZ. 


u\  — 


If  the  focal  length  is  small,  the  magnification  should  be  deter- 
mined with  a  micrometer  microscope.  A  carefully  graduated 
scale  is  a  convenient  object  and  the  size  of  the  image  of  one  or 
more  divisions  is  measured  for  two  positions  of  the  scale  a  known 
distance  apart.  (Principle  of  Abbe's  Focometer.)  Thus  deter- 
mine the  focal  length  of  the  combination  used  in  (A). 

Determine  also  the  focal  length  of  both  the  objective  and  the 
eye-piece  of  a  telescope,  using  the  same  one  as  employed  in  Exp. 
XXXVII,  if  that  has  preceded,  and  calculate  the  magnifying 
power  for  great  distances. 

In  the  report,  draw  careful  figures  of  the  lens  combinations, 
representing  the  principal  foci  and  principal  points  as  calculated 
from  the  final  mean  results. 

(C)  If  the  principal  points  of  a  convergent  system  are  close 
together,  i.  e.,  if  the  lens  or  lenses  may  be  said  to  have  an  optical 
center,  we  may  use  the  following  approximate  method:  If  w  is 
the  distance  between  object  and  image,  and  x  that  between  the 
two  positions  of  the  lens  for  real  images,  u  =  (w  =»=  x)/2,  v  = 
(v=(w=f=x)/2.  Substituting  these  values  in  the  formula  we  get 

J =  • 


MAGNIFYING   POWER   OF   A   TELESCOPE.  133 

If  time  permit,  try  this  method  for  the  combination  of  lenses. 

Questions. 

Describe  a  lens  combination  which  (i)  magnifies  without  distortion;  (2) 
magnifies  without  chromatic  aberration;  (3)  inverts  without  magnifying. 
(See  references.) 

XXXVII   MAGNIFYING  POWER  OF  A  TELESCOPE. 

References—Elementary:  Duff,  §704;  Crew,  §490;  Kimball,  §890;  Reed  &  Guthe, 
§477;  Spinney,  §486; — More  Advanced:  Glazebrook  &  Shaw,  Practical 
Physics,  pp.  358-363. 

The  magnifying  power  of  a  telescope  is  the  ratio  of  the  angle 
subtended  at  the  eye  by  the  image  as  seen  through  the  telescope 
to  the  angle  subtended  by  the  object  viewed  directly.  (If  Exps. 
XXXVI  and  XXXVIII  have  already  been  performed,  use  the 
telescope  employed  in  those  experiments.) 

(A)  Direct  Method. — A  minute  mirror  is  attached  to  the  tele- 
scope by  wax  so  as  to  make  an  angle  of  about  45°  with  the  axis 
and  partly  cover  the  aperture  of  the  eye-piece.  The  telescope  is 
focused  upon  a  scale.  A  second  scale  is  mounted  parallel  to  the 
first  and  near  the  eye-piece,  in  such  a  position  that  the  observer's 
eye  sees,  side  by  side,  the  image  of  the  scale  viewed  through  the 
telescope  and  the  image  of  the  other  scale  reflected  in  the  small 
mirror.  From  the  ratio  of  the  images  of  one  or  more  scale  divi- 
sions, and  their  distances,  the  angles  are  calculated,  and  from 
their  ratio  the  magnifying  power  is  deduced. 

Find  the  magnifying  power  for  at  least  six  distances,  mak- 
ing several  observations  for  each.  Also  determine  the  angular 
field  of  view  of  the  telescope  by  determining  for  each  distance,  r, 
the  total  distance  on  the  scale,  n,  visible  in  the  telescope.  The 
angular  field  of  view,  in  degrees,  will  be  57.3  n/r. 

Indirect  Methods. — The  magnifying  power  defined  above  is 
very  approximately  equal  to  (i)  the  ratio  of  the  magnitude  of 
the  image  to  the  magnitude  of  the  object  when  the  two  are  in  the 
same  plane,  and,  for  great  distances,  is  equal  to  (2)  the  ratio  of 
the  focal  length  of  the  objective  to  the  focal  length  of  the  eye- 
piece. 

The  eye-piece  is  of  such  short  focus  that  the  angle  subtended  by  the  image 
is  practically  the  same  as  if  the  image  were  at  infinity.  For  convenience  we 


134  LIGHT. 

will  consider  the  virtual  image  P'  Q'  (see  Fig.  40)  produced  by  the  eye-piece 
to  be  at  the  same  distance  as  the  object  P  Q. 

Since  the  telescope  usually  views  objects  at  distances  great  compared  with 
its  own  length,  the  angle  subtended  by  the  object  viewed  directly  is  practically 
p  a  Q=p  a  q,  and  the  subtended  by  the  image  is  P'  b  Q'  =p  b  q.  The  ratio  of 
these  two  angles,  which  may  be  taken  as  the  ratio  of  the  tangents,  since  the 
angles  are  small,  =a  c+c  &  =  the  ratio  of  the  focal  length  of  the  objective  to  the 
focal  length  of  the  eye-piece;  and  also,  since  the  length  of  the  telescope  is  short 
compared  with  the  distance  of  the  object,  this  ratio  =Pr  Q'+P  Q,  or  the  ratio 
of  the  magnitude  of  the  image  to  the  magnitude  of  the  object. 


FIG.  40. 

(B)  The  first  approximate  statement  of  the  magnifying  power 
furnishes  the  second  method  for  determining  the  magnifying  power 
for  different  distances  of  the  object.    The  telescope  is  directed 
toward  a  horizontal  scale.     The  scale  is  viewed  through  the  tele- 
scope with  one  eye  and  is  also  observed  with  the  other  eye  by 
looking  along  the  outside  of  the  telescope.     The  eye-piece  is 
moved  in  or  out  until  the  image  appears  at  the  same  distance  as 
the  scale  as  viewed  outside  the  telescope  with  the  other  eye,  i.  e., 
until  there  is  no  parallax  between  the  scale  and  its  image  (no 
relative  motion  of  the  two  as  the  eye  is  moved  about).     It  may 
require  some  practice  to  secure  this.     Determine  the  number  of 
divisions  on  the  scale  which,  as  viewed  directly,  are  covered  by 
the  image  of  one  or  two  large  divisions  as  viewed  through  the 
telescope.    If  it  is  difficult  to  read  the  division  on  the  scale  viewed 
directly,  two  black  strips  may  be  moved  along  the  scale  until 
they  include  the  image  of  one  or  more  divisions  as  seen  through 
the  telescope,  and  the  distance  between  these  strips  read  off. 
Repeat  the  measurements  of  (A). 

(C)  The  second  method  of  defining  the  magnifying  power  of 
a  telescope  is  useful  in  determining  the  magnifying  power  for  very 
distant  objects.    Focus  the  telescope  on  some  very  distant  object. 


RESOLVING   POWER   OF   OPTICAL   INSTRUMENTS.  135 

Without  changing  the  focus,  remove  the  object  glass  and  substi- 
tute for  it  a  diaphragm  with  a  rectangular  opening.  The  ratio 
of  the  focal  length  of  the  objective  to  the  focal  length  of  the  eye- 
piece is  the  ratio  of  a  linear  dimension  of  the  aperture  of  the  dia- 
phragm, L,  to  the  corresponding  dimension,  /,  of  the  image  of 
this  aperture  produced  by  the  eye-piece. 

Since  the  telescope  was  focused  for  parallel  rays,  the  distance,  u,  of  the 
object,  L,  from  the  eye-piece  is  numerically  very  nearly  the  sum  of  the  focal 
lengths,  F+f  (Fig.  41).  /.  the  distance  of  the  image,  /,  formed  by  the  eye- 
piece, is  determined  by 

"I      ,      ^ 


f(F+f)        f(F+f)' 
Hence, 


To  measure  /,  a  micrometer  microscope  (see  p.  13)  may  be 
used,  the  microscope  being  in  line  with  the  axis  of  the  telescope 
and  focused  upon  the  real  image  in  space.  L  may  be  measured 


FIG.  41. 

with  venier  calipers,  or  the  same  micrometer  microscope  may  be 
placed  opposite  the  other  end  of  the  telescope  and  L  measured 
in  the  same  way  as  /. 

Questions. 

1.  Explain  why  the  magnifying  power  should  vary  as  you  have  found  it 
to  do  with  the  distance  of  the  object. 

2.  Which  is  preferable — to  gain  magnifying  power  by  increasing  the  focal 
length  of  the  objective,  or  by  decreasing  the  focal  length  of  the  eye-piece? 
Explain. 

XXXVIII.   RESOLVING  POWER  OF  OPTICAL 
INSTRUMENTS. 

References — Elementary:  Duff,  §695;  Ames,  pp.  483-487. — More  Advanced: 
Drude's  Optics,  pp.  235-236;  Hastings  Light,  pp.  70-72;  Mann's  Advanced 
Optics,  pp.  11-18;  Wood's  Physical  Optics,  pp.  238;  Watson  (Pr.),  §134. 

The  magnification  obtained  with  an  optical  instrument  de- 
pends upon  the  focal  lengths  of  its  lenses,  as  has  been  see  in  the 


136  LIGHT. 

case  of  the  telescope.  The  ability  to  distinguish  details  of  the 
image,  i.  e..,  the  "resolving  power,"  depends  on  the  diameter 
of  the  aperture  through  which  light  enters  the  instrument. 

If  d  is  the  distance  between  two  details  of  an  object  at  a  dis- 
tance D  from  an  aperture  whose  width  parallel  to  these  details 
is  a,  they  may  be  distinguished  if 


^ 
Da 

where  /  is  the  wave-length  of  the  light  employed.    If  the  aperture 
is  circular,  the  equation  is 

d=       I 


where  a  is  the  diameter  of  the  aperture. 

(A)  Resolving  Power  of  Telescope.  —  Metal  gauze  answers  as  a 
very  satisfactory  object  for  studying  the  resolving  power,  as  it 
gives  a  great  amount  of  uniform  detail.     Since  this  detail  con- 
sists of   rectangular  lines,  and  the  aperture  of  the  object  glass 
is  circular,  the  determination  of  the  maximum  distance  at  which 
the  lines  are  discernible  will  be  more  definite,  if  the  aperture  is 
made  rectangular  by  placing  a  slit  in  front  of  the  object  glass. 

Determine  carefully  by  several  settings,  the  maximum  distance, 
D,  at  which  the  lines  of  the  gauze,  parallel  to  the  slit,  are  per- 
ceived. The  gauze  should  be  illuminated  from  behind  by  mono- 
chromatic light  (p.  1  1  6).  Find  carefully  the  mean  distance,  d, 
between  the  centers  of  the  adjacent  wires  of  the  gauze,  and  the 
width  of  the  slit,  a.  Compare  d/D  with  I/a;  I  may  be  obtained 
from  Table  XVIII.  Repeat  with  other  slits  and  other  gauzes. 

(B)  Resolving  Power  of  Eye.  —  With    Porter's   apparatus   the 
resolving  power  of  the  eye  may  be  determined  for  various  aper- 
tures. 

Four  different  gauzes,  37.6,  27,  20,  and  14,  meshes  to  the  cm., 
respectively,  may  be  viewed  through  four  different  apertures  of 
diameters  1  .00  mm.^o.65  mm.,  0.5.3  mm.,  and  0.35  mm.,  respec- 
tively. The  resolving  power  is  determined  by  finding  the  distance, 
D,  from  a  hole  of  diameter  a  to  a  gauze,  of  which  the  distance  be- 


WAVE-LENGTH   OF   LIGHT   BY   DIFFRACTION   GRATING.        137 

• 

tween  the  centers  of  two  adjacent  wires  is  d,  when  the  wires  are 
separately  discernible.  From  the  mean  position  of  several  settings 
of  a  particular  gauze  for  a  particular  aperture,  d/D  should  be  cal- 
culated and  compared  with  1 .2  I/a.  If  ordinary  light  is  used,  / 
may  be  taken  as  0.00006  cm.  Use  each  aperture  and  gauze  in 
succession. 

Questions. 

(1)  Upon  what  does  the  illumination  of  the  image  of  an  optical  instrument 
depend? 

(2)  When  the  diameter  of  the  pupil  of  the  eye  is  4  mm.,  how  far  away  may 
two  points  be  distinguished  which  are  0.2  mm.  apart? 


XXXIX.    WAVE-LENGTH  OF  LIGHT  BY  DIFFRACTION 

GRATING. 

References — Elementary:  Duff,  §696;  Ames,  pp.  530—537;  Crew,  §§479-481,  503; 
Edser  (Light),  pp.  448-458;  Kimball,  §§935~936;  Reed  &  Guthe,  §§494~495; 
Spinney,  §§506-507;  Watson,  §374. — More  Advanced:  Baly's  Spectroscopy, 
Chaps.  VI  and  VII;  Mann's  Advanced  Optics,  Chap.  IX;  Wood's  Physical 
Optics,  pp.  204-216. 

A  diffraction  grating  consists  of  a  great  many  lines  ruled  par- 
allel and  equidistant  on  a  plane  (or  concave)  surface.  If  the  sur- 
face be  that  of  glass,  the  grating  is  a  transmission  grating;  if 
of  metal,  a  reflection  grating.  If  a  transmission  grating  be  placed 
perpendicular  to  homogeneous  parallel  light  from  a  collimator 
(see  Exp.  XXXIII)  and  with  the  lines  parallel  to  the  slit,  a 
series  of  spectra  will  be  formed  on  either  side  of  the  beam  of 
light  transmitted  without  deviation.  If  n  be  the  number  or  order 
of  a  particular  spectrum  counting  from  the  center,  6  the  devia- 
tion or  angle  that  the  rays  forming  the  spectrum  make  with  the 
original  direction  of  the  light,  a  the  grating  space  or  average 
distance  between  the  centers  of  adjacent  lines,  and  /  the  wave- 
length of  the  light 

l  =  -a  sin  d. 

n 

The  deviation  6  is  observed  with  a  spectrometer.  The  grating 
should  be  mounted  in  the  center  of  the  platform  of  the  spectro- 
meter, parallel  to  the  line  joining  two  of  the  leveling  screws, 


138 


LIGHT. 


and  adjusted  as  follows.  (A)  The  lines  of  the  grating  are  per- 
pendicular to  the  plane  of  telescope  and  collimator  when  the 
spectra  are  at  the  same  height  on  each  side.  This  adjustment  is 
secured  by  moving  the  grating  in  its  own  plane  by  turning  simul- 
taneously equal  amounts  in  opposite  direction,  the  above  two 
leveling  screws.  (B)  The  grating  and  telescope  are  next  turned 
to  observe  the  beam  reflected  from  the  grating  as  a  plane  mir- 
ror and  the  image  is  brought  by  the  third  leveling  screw  to  the 
same  height  as  the  direct  ray  when  telescope  and  collimator  are 
in  line.  Finally,  (C)  the  slit  is  opened  wide  and  brightly  illu- 
minated, and  the  grating  turned  until  the  image  reflected  back  on 
the  collimator  lens  from  the  grating,  as  seen  by  looking  just  above 
the  grating,  is  in  the  center  of  the  collimator 
lens.  The  lines  of  the  grating  are  now 
perpendicular  to  the  plane  of  telescope  and 
collimator  and  the  plane  of  the  grating  is 
perpendicular  to  the  axis  of  the  collimator. 
Determine  first  the  wave-length  of  so- 
dium light.  For  a  final  measurement  of 
the  deviation  of  any  spectrum  the  mean  of 
at  least  three  measurements  on  each  side 
should  be  taken.  The  deviations  of  all  the 
spectra  clearly  visible  should  be  obtained. 
If  the  grating  space  be  not  too  small  it 
may  be  obtained  by  measurements  on  a 
dividing  engine  (p.  15),  or  w^ith  a  micrometer 
In  determining  the  grating  space  with  the 
dividing  engine,  secure  the  best  possible  illumination  of  the  lines 
and  focus  the  microscope  until  there  is  no  parallax  when  either 
the  eye  or  the  source  of  light  is  moved.  Set  the  cross-hair  of  the 
microscope  on  a  line  and  read  the  position  of  the  divided  head 
(circular  sale).  Watching  the  lines  through  the  microscope, 
turn  the  screw,  always  in  the  same  direction,  until,  for  example, 
the  tenth  line  is  under  the  cross-hair,  and  read  the  circular  scale. 
Then  turn  the  screw  until  the  tenth  line  from  this  is  under  the 
cross-hair,  read  the  scale,  and  so  on  until  ten  readings 
covering  100  consecutive  lines  have  been  obtained.  In  cal- 
culating subtract  the  first  reading  from  the  6th,  the  2nd 


FIG.  42. 


microscope  (p.  13). 


INTERFEROMETER.  139 

from  the  yth,  etc.,  thus  obtaining  five  values  for  the  space 
occupied  by  50  lines.  From  these  the  width  of  a  single  space  and 
its  possible  error  can  be  found.  Take  ten  such  groups  in  differ- 
ent parts  of  the  grating.  Find  the  average  grating  space  from 
the  mean.  When  the  grating  space  is  very  small,  the  wave-length 
of  some  well-known  spectrum  (e.  g.,  sodium)  is  assumed  in  order 
that  the  grating  space  may  be  derived  by  reversing  the  process 
of  finding  the  wave-length. 

If  time  permit,  determine  the  wave-length  of  as  many  other 
lights  (colors)  as  possible  (see  p.  116). 

Questions. 

1.  What  do  you  observe  as  regards  the  width  of  spectra  of  different  orders? 
What  would  this  indicate  as  regards  the  dispersion  if  mixed  light  or  light  from 
an  incandescent  solid  were  used? 

2.  Compare  the  spectrum  produced  by  a  grating  with  that  produced  by  a 
prism  as  respects,  "rationality,"   "normality,"  brilliancy,  and  dispersion  in 
different  parts  of  the  spectrum. 

3.  What  factor  entering  into  the  final  result  was  most  inaccurate?     How 
might  its  accuracy  be  increased? 

4.  Calculate  the  frequency  of  the  vibrations  of  the  light  employed. 

XL.  INTERFEROMETER. 

References — Elementary:  Duff,  §711;  Crew,  §§493-496;  Kimball,  §926;  Reed  & 
Guthe,  §492;  Watson,  §378. — More  Advanced:  Mann's  Advanced  Optics, 
Chap.  V;  Michelson,  Light  waves  and  Their  Uses;  Wood's  Physical  Optics, 
Chap.  VIII. 

The  interferometer  is  an  instrument  for  determining  the  num- 
ber of  wave-lengths  of  a  monochromatic  light  contained  in  a 

given  distance.     For  a  descrip-  | | 

tion  of  the  interferometer  and  the 
adjustments,  see  the  references. 

The  interferometer  will  be  used 
to  determine  the  wave-length  of 
sodium  light,  assuming  a  knowl- 
edge of  the  true  pitch  of  the  screw. 
This  will  illustrate  the  more  prac- 
tical and  common,  but  also  more 
difficult,  utilization  of  the  in-  FlG 

terferometer    in    determining    a 

length,  assuming  a  knowledge  of  the  wave-length  of  the  light 
employed. 

The  light  5  (see  figure)  had  best  be  monochromatic,  e.  g.,  a 


y 


140  LIGHT. 

sodium  flame.  Initially  place  the  mirror  D  at  approximately  the 
same  distance  from  the  rear  face  of  A  as  the  distance  from  this 
surface  to  C.  Adjust  C  until  its  image  coincides  with  either  of 
the  images  from  D.  (There  will  be  two  images  owing  to  reflection 
from  the  two  faces  of  A.)  A  slight  adjustment  will  now  give  the 
fringes  (alternate  light  and  dark  bands,  preferably  arcs  of  circles) . 
The  observer  must  look  at  A  in  a  direction  parallel  to  AD. 

Move  the  mirror  D  by  means  of  the  worm,  and  count  the  num- 
ber of  fringes  which  pass  over  the  field  of  view.  A  needle  in  front 
of  A  may  help  as  an  index. 

From  the  number  of  turns  and  fractional  turns  of  the  screw 
and  the  value  of  the  pitch  of  the  screw,  find  the  distance  D  has 
moved  and  from  this  and  the  number  of  fringes  which  have  passed, 
calculate  the  wave-length.  Notice  that  the  length  of  the 
path  of  the  light  changes  by  twice  the  displacement  of  the  mir- 
ror D. 


XLI.    ROTATION  OF  PLANE  OF  POLARIZATION. 

References — Elementary:  Duff,  §§752-753,  760-762;  Ames,  pp.  563-565;  Edser 
(Light),  pp.  503-509;  Kimball,  §958;  Reed  &  Guthe,  §520;  Spinney,  §542; 
Watson,  §§407-413. — More  Advanced:  Ewell,  Physical  Chemistry,  pp.  217- 
223;  Mann's  Advanced  Optics,  Chap.  XII;  Watson's  (Pr.)§§i48-i5i;  Wood, 
Physical  Optics,  Chap.  XVII. 

Plane  polarized  light  is  obtained  by  passing  light  through  a 
Nicol's  prism.  If  the  light  be  then  allowed  to  fall  on  a  second 
Nicol's  prism  that  can  be  rotated,  there  will  be  two  positions  of 
this  second  prism  in  a  complete  rotation  in  which  no  light  will 
pass  through.  If  an  optically  active  substance,  such  as  a  solu- 
of  cane  sugar,  be  then  introduced  between  the  two  Nicols,  it 
will  rotate  the  plane  of  polarization  of  the  light  which  falls  on 
the  second  prism,  and  then,  to  quench  the  light,  the  second 
prism  must  be  rotated  through  an  equal  angle.  Thus  the  rota- 
tion produced  by  the  sugar  is  measured. 

The  specific  rotation  is  defined  as  the  rotation  per  decimeter, 
per  gram  of  active  substance  in  one  cubic  centimeter.  If  <p  is  the 
angle  observed  at  t°  with  yellow  (D)  light,  /  the  length  of  the 
solution  in  decimeters,  m^  the  mass  of  the  solvent,  m  the  mass 


ROTATION   OF   PLANE   OF   POLARIZATION.  14! 

of  the  solute,  p  the  density  of  the  solution  and  [<p]  *D  the  speci- 
fic rotation,  the  volume  of  the  solution  is  (WQ  -j-  w)/p  and 


Monochromatic  light  must  be  used  and  a  sodium  flame  is 
most  convenient  (see  p.  1  1  6).  The  light  rays  must  be  made  par- 
allel before  they  fall  on  the  polarizing  prism,  otherwise  rays  in 
different  directions  would  pass  through  different  thicknesses  of 
the  sugar  and  would  consequently  be  rotated  by  different 
amounts.  Parallel  light  may  be  obtained  by  putting  the  source 
at  the  principal  focus  of  a  convex  lens  through  which  the  light 
must  pass  before  falling  on  the  polarizing  Nicol.  The  light 
must  also  be  parallel  to  the  axis  of  the  Nicol. 

The  empty  tube  intended  to  contain  the  sugar  solution  is 
hrst  placed  in  position  between  the  prisms  and  the  position  of 
the  analyzing  Nicol  noted,  on  the  circular  scale,  when  the  light 
is  quenched.  This  setting  will  be  facilitated  by  using  a  screen  to 
cover  all  but  a  small  central  part  of  the  prism.  It  may  be  found 
that  the  Nicol  can  be  rotated  through  an  appreciable  angle  with- 
out the  light  reappearing.  The  best  that  can  be  done  is  to  take 
the  middle  of  this  space  as  the  position  of  extinction.  The  obser- 
vation should  be  repeated  a  number  of  times  and  the  mean  taken. 
The  analyzing  Nicol  should  then  be  rotated  through  180°  and 
the  readings  repeated.  Sugar  solutions  of  different  strengths 
(which  should  be  carefully  made  up  and  recorded)  are  then  intro- 
duced in  succession  into  the  tube  and  the  rotations  they  produce 
observed.  The  zero  readings  should  be  frequently  repeated. 
The  length  of  the  tube  should  also  be  obtained,  so  that  the  rota- 
tion per  decimeter  may  be  deduced.  With  the  results  obtained, 
a  curve  should  be  plotted,  specific  rotations  being  ordinates  and 
concentrations  abscissae. 

The  apparatus  above  described  is  simple  but  very  imperfect 
in  its  action.  The  sensitiveness  is  greatly  increased  by  introduc- 
ing between  the  polarizer  and  the  specimen  a  so-called  biquartz, 
two  parallel,  abutting  plates  of  quartz,  one  with  left  rotary  power 
and  the  other  with  right.  A  source  of  white  light  must  be  em- 
ployed, for  example,  a  frosted  incandescent  bulb,  and  the  analyzer 


142 


LIGHT. 


is  set  for  equality  of  color  in  the  two  halves.    There  are  two 
common  colors,  but  the  darker  is  preferable. 

Fig.  44  explains  the  color  changes.  R  and  L  are  the  two  halves  of  the 
biquartz,  viewed  from  the  analyzer.  The  two  halves  are  of  such  a  thickness 
(3-75  mm.)  that  the  plane  of  polarization  of  yellow  light  is  rotated  through  90°. 
Owing  to  the  rotatory  dispersion  the  other  colors  will  be  rotated  different 
amounts  as  shown  by  the  letters  R  (red)  and  B  (blue).  If  the  analyser  is  set 
to  transmit  light  vibrations  parallel  to  those  which  left  the  polarizer,  the  yellow 

light  will  be  omitted  and  each  half  of  the  bi- 
quartz will  appear  of  a  purplish  color  ("tint  of 
passage").  If  the  analyser  is  displaced  slightly 
clockwise,  more  of  the  red  component  on  the 
right  will  be  transmitted  and  less  of  the  blue, 
and  therefore  this  half  will  appear  red  and  the 
other  half  will  appear  blue. 

If  a  dextrorotatory  specimen  is  placed  be- 
tween the  biquartz  and  the  analyzer  the  direc- 
tions of  vibration  of  the  different  colors  will  be 
rotated  to  the  positions  indicated  by  the  dotted 
lines  and  the  analyzer  must  be  rotated  to  a  new 
position  (/'),  perpendicular  to  the  emerging  yel- 
low vibration,  in  order  to  have  the  two  halves 
the  same  color.  With  the  help  of  the  biquartz 
the  analyzer  can  be  set  within  about  a  tenth  of 
a  degree. 

The  effective  thickness  of  the  biquartz 
will  not  be  correct  (90  degrees  rotation 
of  sodium  light),  unless  it  is  perpendic- 
ular to  the  axis  of  the  Nicol's  prisms. 
This  may  be  secured  by  using  sodium 
light  and  analyzer  set  for  extinction,  and  then  placing  the  biquartz 
in  such  a  position  that  there  is  still  extinction  when  the  analyzer 
is  rotated  90°. 


FIG.  44. 


*  (W 

L^t  Polarizer 


Lens 


Analyzer  Eyepiece 


A  still  more  sensitive  instrument  is  the  Lippisch  half-shade 
polarimeter,  the  essential  parts  of  which  are  represented  in 
horizontal  section  in  Fig.  46.  The  smaller  Nicol's  prism  of 
the  polarizer  is  at  a  small  angle  with  the  larger  Nicol's  prism, 
so  that,  unless  the  analyzer  is  properly  placed,  parallel  or  per- 
pendicular to  the  bisector  of  the  angle  between  them,  the  two 


ROTATION   OF   PLANE   OF   POLARIZATION.  143 

parts  of  the  field  appear  of  different  shades.  The  analyzer  is  set 
so  that  both  halves  of  the  field  appear  equally  dark,  first  with- 
out the  specimen,  then  with  the  specimen,  and  the  difference  is 
the  angle  of  rotation.  The  angle  between  the  two  Nicol's  prisms 
of  the  polarizer  can  be  varied.  The  smaller  the  angle,  the  greater 
the  sensitiveness  but  the  illumination  is  thereby  reduced. 

In  addition  to  the  lens  or  lenses  for  rendering  the  incident  light 
parallel,  sensitive  polarimeters  are  commonly  provided  with  an 
eye-piece  which  is  focused  upon  the  biquartz  or  other  contrast 
field.  If  the  zero  readings  are  taken  with  a  tube  filled  with 
water,  less  change  of  focus  will  be  required  upon  introducing 
the  active  substance,  than  if  the  zero  readings  had  been  taken 
with  simply  air. 


r 


Lens 

FIG.  46. 

In  all  polarimeters  there  are  obviously  two  positions  of  the 
analyzer,  180°  apart  when  the  two  halves  appear  equally  dark 
or  of  the  same  dark  shade.  There  are  also  two  intermediate 
positions  where  the  two  halves  appear  of  the  same  bright  color 
or  equally  bright.  The  latter  positions  are  not  as  sensitive  as  the 
former.  i 

Questions. 

1.  How  can  the  rotation  be  partially  explained?     (See  references.) 

2.  What  is  the  chemical  characteristic  of  substances  that  are  optically 
active  in  solution? 

3.  Wherein  does  the  rotation  produced  by  a  solution  differ  from  that  pro- 
duced by  a  magnetic  field? 

4.  What  would  be  the  effect  of  using  white  light  in  the  first  part  of  the  experi- 
ment? 


ELECTRICITY  AND  MAGNETISM. 

29.  Resistance-boxes. 

A  resistance-box  consists  of  a  number  of  resistance  coils  joined 
so  that  each  one  bridges  the  gap  between  two  of  a  series  of  brass 
blocks  placed  in  line  on  the  cover  of  the  box  within  which  the 
coils  are  suspended.  For  each  gap  a  plug  or  connector  is  also 
provided,  and  when  the  plug  is  inserted  into  the  gap  the  resist- 
ance at  the  gap  is  "cut  out"  or  practically  reduced  to  zero.  The 
coils  are  wound  so  as  to  be  free  from  self-induction.  The  successive 
resistances  are  arranged  in  the  same  order,  and  are  of  the  same 
relative  magnitudes  as  the  successive  weights  in  a  box  of  weights. 
By  removing  the  proper  plugs  any  combination  of  resistances  can 
be  obtained  from  the  smallest  to  the  sum  of  all.  Before  beginning 
work,  it  is  advisable  to  clean  the  plugs  with  fine  emery-cloth 
so  that  they  may  make  good  contacts,  and  thereafter  care  should 
be  taken  not  to  soil  them  with  the  fingers. 

One  important  precaution  in  regard  to  the  use  of  the  resistance- 
box  should  be  observed.  If  any  of  the  plugs  are  in  loosely,  there 
will  be  some  resistance  at  the  contact.  Hence,  the  plug  should  be 
screwed  in  firmly,  but  not  violently.  When  any  one  plug  has 
been  withdrawn,  the  others  should  be  tested  before  proceeding, 
for  the  removal  of  one  may  loosen  the  contact  of  the  others. 
This  precaution  is  especially  important  in  making  a  final  deter- 
mination. 

30.  Forms  of  Wheatstone's  Bridge. 

The  practical  measurement  of  a  resistance  consists  in  comparing 
it  with  a  known  or  standard  resistance.  Wheatstone's  Bridge  is 
an  arrangement  of  conductors  for  facilitating  this  comparison, 
and  consists  essentially  of  six  branches  which  may  be  represented 
by  the  sides  and  diagonals  of  a  parallelogram  (see  Fig.  47).  The 
unknown  resistance,  R,  and  the  known  resistance,  S,  form  two 
adjacent  sides.  The  other  two  sides  are  formed  by  two  conduc- 

144 


GALVANOMETERS . 


145 


tors  of  resistance  P  and  Q,  which,  however,  do  not  need  to- be 
known  separately,  provided  their  ratio  be  known.  One  of  the 
diagonals  contains  a  battery  and  the  other  a  galvanometer.  If 
the  ratio  of  P  to  Q  is  adjusted  until  no  current  flows  through 
the  galvanometer,  R:S::P:Q.  (See  references  under  Exp.  XLIV.) 

Two   forms  of  the   Wheatstone   Bridge   arrangement   are   in 
common  use.    One  is  called  the  Wire  (or  meter)  Bridge;  the  other, 
which  uses  a  box  of  adjusted  re- 
sistances, is  called  a  Bridge  Box. 
In  the  wire   bridge   the    "ratio 
arms"  (whose  resistances  are  P 
and  Q)  are  the  two  parts  of  a  uni- 
form wire  i  meter  long,  and  the 
ratio  of  P  to  Q  is  that  of  the 
lengths  of  the  corresponding  parts    „ 
of  the  wire.    The  known  resist- 
ance, 5,  may  be  that  of  a  standard 
coil  or  one  of  the  known  resist- 
ances of  a  resistance-box. 

The  Bridge  Box,  or  "Post- 
office  Bridge,"  consists  of  a  resist- 
ance-box with  three  series  of  resistances  in  line,  forming  three 
arms  of  the  Wheatstone  Bridge,  the  unknown  resistance  forming 
the  fourth  arm.  The  "ratio  arms"  consist  of  resistances  of 
i,  10,  100,  1000  (all  of  which  are  not  always  necessary),  so  that 
the  calculation  of  the  ratio  is  very  simple.  Keys  for  closing  the 
battery  and  galvanometer  branches  are  also  usually  mounted 
on  the  box. 

31.  Galvanometers. 

References — Northrup,  Journal  of  the  Franklin  Institute  (1910),  p.  295;  Watson 

(Pr.}>  §§170-174- 

Types  of  Galvanometers. 

There  are  two  chief  types  of  reflecting  galvanometers.  In 
both  the  principle  at  basis  is  that  if  a  magnet  be  placed  in  the 
plane  of  a  coil  of  insulated  wire,  on  passing  a  current  through  the 
coil  both  magnet  and  coil  become  subject  to  forces  that  tend 
to  set  them  at  right  angles  to  each  other.  In  the  Thomson  type 
10 


FIG.  47. 


146  ELECTRICITY   AND   MAGNETISM. 

the- coil  is  fixed  and  the  magnet  suspended  within  the  coil  is  free 
to  turn,  while  in  the  d'Arsonval  type  the  magnet,  of  a  horseshoe 
form,  is  fixed,  and  the  coil,  suspended  between  the  poles  of  the 
magnet,  is  free  to  turn. 

The  sensitiveness  of  the  Thomson  galvanometer  is  greatly  in- 
creased in  two  ways:  first,  two  magnetic  needles,  forming  an 
astatic  pair,  are  attached  to  the  same  axis  of  rotation,  second,  an 
external  control  magnet  is  used  to  weaken  the  restraint  of  the 
earth's  magnetic  force  or  even  to  overcome  the  earth's  field  and 
produce  a  suitable  field  of  its  own.  The  chief  difficulty  in  greatly 
increasing  the  sensitiveness  by  means  of  the  control  magnet  is 
that  slight  variations  of  the  whole  magnetic  field,  due  to  outside 
currents  or  movements  of  magnetic  materials  in  or  near  the  lab- 
oratory, disturb  the  needle. 

On  the  d'Arsonval  galvanometer  variations  of  the  external  mag- 
netic field  have  practically  no  effect,  since  its  own  magnetic  field 
is  very  strong.  On  the  other  hand,  the  torsion  of  the  fine  suspend- 
ing wire  through  which  the  current  has  to  pass  changes  somewhat 
with  the  temperature,  so  that  the  zero  reading  of  the  galvano- 
meter is  subject  to  some  change.  The  sensitiveness  can  be  in- 
creased by  increasing  the  strength  of  the  magnet,  but  there  is  a 
limit  to  this,  since  small  traces  of  iron  are  always  present  in  the 
wire  and  insulation  of  the  coil,  and  this,  acted  on  by  the  magnetic 
field,  exercises  a  magnetic  control  that  is  proportional  to  the 
square  of  the  strength  of  the  field.  When  an  extremely  sensitive 
galvanometer  for  very  accurate  work  is  required,  the  Thomson 
type  must  be  used. 

Classes  of  Galvanometers. 

Galvanometers  may  also  be  classified  according  to  the  purpose 
for  which  they  are  particularly  suited. 

(1)  Detector  Galvanometers  or  galvanoscopes,  which  are  used 
to  ascertain  whether  a  current,  steady  or  temporary  exists.    Any 
of  the  following  may  also  be  used  for  this  purpose. 

(2)  Current  Galvanometers  which  serve  to  measure  steady  cur- 
rents by  deflections  proportional  to  the  currents,  or  to  measure 
e.  m.  f's  assumed  (on  account  of  the  high  resistance  of  the  galvan- 
ometer) to  be  proportional  to  the  currents. 


GALVANOMETERS.  147 

(3)  Quantity  Galvanometers  which  are  used  to  measure  quan- 
tities of  electricity  by  deflections  proportional  to  the  quantities, 
in  this  class  we  may  distinguish  the  following: 

(a)  Ballistic  Galvanometers  in  which  the  damping  is  reduced 
to  a  minimum  and  which  are  so  called  from  the  analogy  of  the 
ballistic  pendulum.    These  might  also  be  called  undamped  quan- 
tity galvanometers. 

(b)  Flux  galvanometers,  or  fluxmeters  in  which  the  electro- 
magnetic damping  (due  to  currents  induced  by  the  motion  of  the 
coil)  is  very  great  and  which  are  so  called  because  they  are  par- 
ticularly useful  for  measuring  quantities  of  electricity  due  to 
changes  of  magnetic  flux  through  some  part  of  the  circuit.    They 
might  also  be  called  damped  quantity  galvanometers. 

A  detector  galvanometer  may  be  of  either  type  and  need  not  be 
carefully  constructed  or  adjusted  so  as  to  give  deflections  propor- 
tional to  the  current  or  to  the  quantity  of  electricity  that  passes. 

A  current  galvanometer  should  theoretically  be  constructed 
so  that  the  deflection  to  the  full  extent  of  the  scale  is  proportional 
to  the  current.  It  is,  for  two  reasons,  difficult  to  satisfy  this 
condition:  (i)  the  deflections  are  read  on  a  straight  scale,  (2) 
the  magnetic  field  in  which  the  swinging  system  moves  is  not 
uniform,  (i)  Applies  to  galvanometers  of  either  type  and  in- 
creases rapidly  with  the  deflection.  It  is  readily  calculated 
(from  trigonometrical  tables)  that  if  the  currents  are  actually 
as  2 :  i  and  the  smaller  angular  deflection  is  B  =  5°,  the  ratio  of 
the  deflections  on  the  linear  scale  is  2.05,  so  that  the  error  is 
2.5%  while  if  B  =  3°,  the  error  is  less  than  i%.  In  the  first  case 
the  larger  deflection  on  the  linear  scale  (at  a  distance  of  50  cm.) 
would  be  about  18  cm.  and  in  the  second  case  about  9  cm.  (2) 
Does  not  apply  seriously  to  the  Thomson  galvanometer  since 
the  needle  is  very  small  and  the  part  of  the  field  in  which  it  moves 
is  sensibly  uniform ;  but  it  may  cause  very  large  errors  in  the  case 
of  a  d'Arsonval. galvanometer  with  a  large  coil.  The  difficulty 
is  sometimes  partly  overcome  by  curving  the  faces  of  the  mag- 
netic poles  toward  the  coil  so  as  to  make  the  magnetic  field  more 
nearly  uniform;  but  the  only  safe  plan  for  accurate  work  is  to 
calibrate  the  galvanometer  by  applying  e.  m.  f's  in  known  ra- 
tios (see  §35  p.  153). 


148  ELECTRICITY   AND   MAGNETISM. 

A  ballistic  galvanometer  is  a  reflecting  galvanometer  of  either 
type,  so  made  that  its  period  of  swing  is  very  long,  so  that  it  starts 
into  motion  only  very  slowly.  If  this  condition  be  fulfilled,  and 
if  it  be  subject  to  only  very  slight  damping  of  its  motion,  the 
galvanometer  may  be  used  for  comparing  quantities  of  electri- 
city suddenly  discharged  through  the  coils  of  the  galvanometer, 
for,  practically  speaking,  all  the  electricity  will  have  passed  be- 
fore the  swinging  system  has  appreciably  moved  from  this  posi- 
tion of  rest.  In  these  circumstances  it  can  be  shown  that  the 
quantity  of  electricity  is  proportional  to  the  sine  of  half  the  angle 
of  the  first  swing,  or  (if  the  angle  is  very  small)  practically  to  the 
deflection  as  read  on  the  scale. 

A  flux  galvanometer  has  such  high  electromagnetic  damping 
that,  in  comparison  with  it,  the  effects  of  the  air  damping  and 
of  the  torsional  resistance  of  the  suspension  may  be  neglected. 
During  the  change  of  flux  the  applied  e.  m.  f.  acts  against  the 
ohmic  resistance  the  e.  m.  f  .  induced  by  the  motion  of  the  coil  and 
the  e.  m.  f.  of  self-induction  in  the  coil,  or 

dN 


The  pole  pieces  of  the  magnet  are  curved  so  that  the  field  in 
which  the  coil  moves  is  approximately  uniform.  Hence,  from 
dynamical  principles,  if  w  is  the  angular  velocity, 


where  I  is  the  moment  of  inertia  of  the  coil  and  C  is  a  constant. 
Eliminating  i,  we  get 


=  _  CdN_CLdi 

dt"    ~  r    dt     r  dt       r   dt' 

Integrate  between  the  beginning  of  the  motion  and  the  end  of 
the  swing,  noting  that  co  and  i  are  zero  at  both  limits.     Hence 


Two  methods  of  reading  the  deflection  of  a  galvanometer  are  in 
common  use.     In  one,  called  the  English  or  objective  method,  a 


CORRECTION  FOR  DAMPING  OF  A  BALLISTIC  GALVANOMETER.    149 

beam  of  light  reflected  from  the  mirror  of  the  galvanometer  falls 
on  a  scale,  forming  a  spot  of  light  which  moves  as  the  needle  or 
coil  is  deflected.  In  the  other,  called  the  German  or  subjective 
method,  the  image  of  a  scale  formed  by  the  mirror  of  the  galvan- 
ometer is  read  by  a  telescope  with  a  cross-hair. 

Devices  for  Bringing  a  Galvanometer  to  Rest. — For  bringing  to 
rest  the  needle  of  a  ballistic  Thomson  galvanometer  a  coil  is 
mounted  on  the  outside  of  the  galvanometer  in  front  of  the  lower 
needle.  The  terminals  of  the  coil  are  brought  to  a  reversing 
switch  by  which  the  current  from  a  cell  can  be  sent  through  the 
coil  in  either  direction.  By  suitably  choosing  the  direction  and 
duration  of  the  current,  the  needles  and  mirror  may  be  brought 
to  rest.  (A  current  in  this  coil  affects  the  needle  in  the  same 
manner  as  would  a  current  in  one  of  the  regular  galvanometer 
coils,  but  it  is  much  more  convenient  to  use  a  separate  coil  like 
this,  which  is  readily  accessible  and  which  does  not  interfere  with 
the  other  connections.) 

The  suspended  system  of  either  type  of  galvanometer  may 
also  be  brought  to  rest  by  short-circuiting  the  galvanometei  by 
a  simple  key  directly  connected  to  the  terminals.  For,  by  Lenz's 
Law,  the  currents  induced  are  such  as  to  bring  the  moving  coil 
or  needle  to  rest.  If  the  resistance  of  the  coils  is  high,  this  method 
is  slow,  and  the  following  more  rapid  method  may  be  used. 
A  coil  in  which  a  small  bar  magnet  can  be  moved  is  placed  in 
series  with  the  short-circuiting  key.  By  suitably  moving  the 
magnet  in  and  out,  currents  are  induced  which  will  quickly  bring 
the  suspended  system  to  rest. 

32.  Correction  for  Damping  of  a  Ballistic  Galvanometer. 

References — Kohlrauschs  Physical  Measurements,  §51;  Stewart  and  Gee's  Prac- 
tical Physics,  II,  pp.  364-369. 

In  considering  the  throw  proportional  to  the  charge  passing 
thiough  the  coils  of  a  ballistic  galvanometer,  we  assume  that  the 
galvanometer  is  free  from  damping;  i.  e.,  that  the  suspended  sys- 
tem, needles,  mirror,  etc.,  experiences  no  resistance  to  turning. 
Since  this  is  never  realized,  a  correction  must  be  applied  to  the 
throw. 

The  correction  is  not  of  importance  where  we  compare  throws, 


150 


ELECTRICITY   AND   MAGNETISM. 


since  the  correction  cancels  out,  but  in  much  work  with  ballistic 
galvanometers  this  correction  is  very  important. 

Sel  the  needle  vibrating  and  record  n  +  I  successive  turning- 
points.     From  these  we  obtain  by  successive  subtraction  n  suc- 
cessive full  vibrations  of  the  needle  from  one  side  to  the  other. 
Call  the  first  full  vibration  a\  and  the  last  an.    Then  the  correc 
tion  by  which  each  throw  should  be  multiplied  is  (i  -j-  V2)  where 

logai-logan 


jf 


33 •  Galvanometer  Shunts. 

References — Ayrton  &  Mather,  Practical  Electricity,  §§106-111. 

If  a  galvanometer  of  resistance  G  is  shunted  by  a  shunt  of  re- 
sistance 5  and  if  C  is  the  whole  current  and  C\  the  current  through 
the  galvanometer 

Ci=     5 

C     G+S' 

Galvanometers  are  frequently  supplied  with  shunt-boxes  in 
which  the  ratio  of  S:G  are  1/9,  1/99,  1/999,  so  that  the  values 

of  S:(G+S)  are  i/io,  i/ioo,  i/iooo. 
Such  a  shunt-box  cannot  easily  be  used 
with  any  galvanometer  except  that  for 
which  it  was  designed. 

Universal  shunt-boxes  are  also  made 
which  can  be  used  with  any  galvano- 
meter. Such  a  box  consists  of  a  series  of 
high  resistances  connected  as  indicated  in 
the  figure.  AB  is  a  coil,  of  resistance  S, 
connected  to  the  galvanometer,  of  resist- 
ance G.  Let  the  current  through  the  galvanometer  be  d,  and 
let  the  whole  current  be  C.  Then  as  above 

r_    CS 

G+S' 

Now  let  the  battery  circuit  be  connected  to  A  and  P,  where  the 
resistance  of  AP  is  S/n.    Denoting  the  current  through  the  gal- 


VVV\AA/V\AAAA/V 
4 


FIG.  48. 


GALVANOMETER   SHUNTS.  15! 

vanometer  by  C/,  and  the  whole  current  by  C  and  making  the 
proper  changes  in  the  above  equation, 

d'  S/n  iS 


C     S/n+(S-S/n)  +  G    n  S+GJ 
,.<Y-I     * 


nS+G' 

Hence  when  a  current  is  connected  to  A  and  P,  the  galvanometer 
deflection  is  i/n  as  great  as  when  the  same  current  is  connected 
to  A  and  B,  or  the  sensitiveness  is  i/n  as  great.  By  subdividing 
AB,  the  values  of  3,  10,  100,  etc.,  are  given  to  n. 

Shunting  a  Ballistic  Galvanometer. — The  formulae  stated  above 
were  deduced  from  Ohrn's  Law  for  steady  direct  currents.  It 
can,  however,  be  shown  that  shunts  like  the  above  may  be  used 
in  the  same  way  with  ballistic  galvanometers  through  which 
charges  of  electricity  are  passed.  To  prove  this,  all  we  need  to  do 
is  to  show  that  charges,  like  steady  direct  currents,  divide  in  a 
parallel  arc  into  parts  inversely  as  the  ohmic  resistances.  Con- 
sider any  one  of  several  branches  in  a  parallel  arc.  Let  the  part 
of  the  charge  that  passes  through  it  be  q\,  and  let  the  magnitude 
of  the  instantaneous  current  through  it,  at  time  /  after  the  be- 
ginning of  the  discharge,  be  i\.  The  induced  e.  m.  f.  at  that 
moment  is  Lidii/dt  where  LI  is  the  self -inductance  of  the  branch. 
Suppose  the  discharge  is  caused  by  connecting  an  e.  m.  f.  E  to 
the  parallel  arc  for  a  short  time  and  then  disconnecting  it,  and 
let  the  whole  time  of  rise  and  fall  of  the  brief  current  be  T.  Then 


ET 


Hence  the  charges  through  the  various  branches  are  inversely 
as  their  ohmic  resistances.     If  the  above  proof  be  carefully  ex- 


152  ELECTRICITY   AND   MAGNETISM. 

amined,  it  will  be  seen  that  it  simply  means  that  the  total  quan- 
tity due  to  the  induced  e.  m.  f.  is  zero,  since  the  induced  current 
in  the  first  half  of  the  process  is  opposite  to  that  in  the  second 
half. 

34.  Standard  Cells. 

References — Elementary:  Duff,  §473;  Watson,  §§551,  554-555. — More  Advanced: 
Bureau  of  Standards  Bulletin,  Nos.  67,  70,  71,  Circular,  No.  29;  Ewell, 
Physical  Chemistry,  pp.  334-336;  Henderson's  Electricity  and  Magnetism, 
pp.  176-182;  Watson  (Pr.),  §§202-203. 

The  standard  Daniell  cell  consists  of  an  amalgamated  zinc  rod 
dipping  into  a  porous  cup  containing  a  solution  of  sulphate  of 
zinc,  which,  in  turn,  stands  in  a  glass  vessel  containing  a  copper 
sulphate  solution  and  a  copper  plate.  To  amalgamate  the  zinc 
rod,  thoroughly  clean  it  with  sand-paper,  dip  it  in  dilute  sul- 
phuric acid,  and  rub  over  it  a  few  drops  of  mercury  with  a  cloth. 
The  porous  cup  should  be  thoroughly  cleaned  inside  and  out. 
The  copper  plate  should  be  cleaned  bright  with  sand-paper. 
The  porous  cup  is  half-filled  from  a  stock  bottle  with  a  solution 
of  zinc  sulphate  (44.7  g.  of  crystals  of  c.  p.  zinc  sulphate  dissolved 
in  100  c.c.  of  distilled  water).  The  zinc  rod  is  introduced  and 
the  porous  cup  is  placed  in  the  glass  vessel,  which  is  filled,  not 
quite  up  to  the  level  of  the  zinc  sulphate  in  the  porous  cup,  with 
copper  sulphate  solution  (39.4  g.  of  c.  p.  copper  sulphate  dis- 
solved in  100  c.c.  of  distilled  water).  The  copper  plate  is  also 
placed  in  the  outer  vessel.  After  being  set  up,  the  cell  should 
be  short-circuited  for  15  minutes  and  then  allowed  to  stand  on 
an  open  circuit  for  5  minutes.  The  cell  should  not  remain  set  up 
more  than  a  few  hours.  When  it  is  no  longer  needed,  pour  the 
copper  sulphate  solution  back  into  the  stock  bottle  and  the  zinc 
sulphate  solution  back  into  its  bottle,  unless  the  zinc  has  turned 
black,  in  which  case  throw  the  zinc  sulphate  away.  The  e.  m.  f. 
of  the  Daniell  cell,  prepared  as  above,  is  1.105  international  volts, 
correct  Jo  0.2  per  cent. 

The  Clark  cell,  which  differs  from  the  above  in  the  fact  that  the 
copper  is  replaced  by  mercury  and  the  copper  sulphate  by  mer- 
curous  sulphate,  is  a' more  constant  standard  than  the  Daniell 
cell,  but  it  needs  to  be  treated  with  much  greater  care,  since  the 


DOUBLE    COMMUTATOR. 


153 


passage  of  a  very  small  current  through  it  will  alter  the  e.  m.  f. 
Hence  it  can  be  used  only  for  null  methods  and  kept  in  circuit  for 
the  briefest  time  possible.  At  temperature  /  its  e.  m.  f.  in  volts  is 

i.433-.ooi2(/-i5). 

In  the  cadmium  cell,  which  has  been  since  1908  the  international 
standard,  the  zinc  and  zinc  sulphate  of  the  above  are  replaced 
by  cadmium  and  cadmium  sulphate.  Its  e,  m.  f.  is 

1 .01 830  —  .oooo4o6(/  —  20) . 

35.  Device  for  Getting  a  Small  E.  M.  F. 

In  many  experiments  it  is  desirable  to  use  an  e.  m.  f.  much 
smaller  than  that  of  a  single  cell.  To  get  such  an  e.  m.  f.,  a  box 
of  very  high  resistance  may  be  placed  in  series  with  a  constant 
cell  and  any  desired  fraction  of  the  whole  e.  m.  f.  may  be  obtained 


FIG.  49. 

by  tapping  off  from  various  points;  e.  g.,  at  the  ends  of  a  resist- 
ance r  out  of  the  total  resistance  R  of  the  box  (Fig.  49).  The 
e.  m.  f.  thus  obtained  may  be  found  from  Ohm's  Law,  but  it 
must  be  noticed  that  the  resistance  between  the  terminals  of  r 
is  the  resistance  of  a  parallel  arc.  If,  however,  the  resistance  of 
the  branch  circuit  be  proportionally  very  large  and  that  of  the 
cell  proportionally  very  small,  both  may  be  omitted  in  the  cal- 
culation. 

36.  Double  Commutator. 

It  is  sometimes  desirable  to  be  able  to  reverse  two  parts  of  a  net- 
work repeatedly  and  at  the  same  rate.    For  this  purpose  a  double 


154  ELECTRICITY   AND   MAGNETISM. 

commutator  is  convenient.  It  consists  of  two  two-part  commuta- 
tors mounted  on  a  common  shaft;  e.  g.,  on 
opposite  ends  of  the  shaft  of  a  small  motor. 
If,  for  example,  the  battery  used  with  a 
Wheatstone's  Bridge  be  connected  through 
one  commutator  while  the  galvanometer 
is  connected  through  the  other,  an  alter- 

-L         Y(          \  nating  current  will  act  in  the  arms  of  the 

\^        J\  bridge,  while  a  direct  current  (or  a  suc- 

I <<^—^  L. cession  of  unidirectional  pulses)  will  pass 

FIG.  50.  through  the  galvanometer. 

37.  Relation  Between  Electrical  Units. 

(E.S.  =  Electrostatic;  E.M.  =  Electromagnetic.) 

Ampere        =  lo-1   C.G.S.-E.M.  units  of  current. 

Coulomb      =  ro-1  C.G.S.-E.M.  units  of  quantity. 

Volt  =  io8    C.G.S.-E.M.  units  of  electromotive  force. 

Ohm  =  io9    C.G.S.-E.M.  units  of  resistance. 

Farad  =  io-9  C.G.S>E.M.  units  of  capacity. 

Microfarad  =  io-15  C.G.S.-E.M.  units  of  capacity. 

Henry  =  io9   C.G.S.-E.M.  units  of  indurcance. 

Volt  =i^xio-2  C.G.S.-E.S.  units  of  electromotive  force.0 

Coulomb      =  3  X  io9   C.G.S.-E.S.  units  of  quantity. 

Microfarad  =  9Xio5   C.G.S.-E.S.  units  of  capacity. 


XLII.    HORIZONTAL  COMPONENT  OF  EARTH'S 
MAGNETIC  FIELD. 

References— Elementary:  Duff,  §§377-384;  Ames,  pp.  609-613;  Crew,  §§306- 
308;  Hadley,  Chap.  VI;  Kimball,  §§491,  501;  Reed  &  Guthe,  §236;  Spinney, 
§§305-3o8;  Watson,  §§424-429. — More  Advanced:  Kohlrausch's  Physical 
Measurements,  pp.  240-247;  Stewart  &  Gee's  Practical  Physics,  pp.  284-309; 
Watson  (Pr.),  §§164-166. 

In  this  experiment  the  horizontal  component  of  the  earth's 
magnetic  field,  at  a  point  in  the  laboratory,  is  deduced  from  the 
period  of  vibration  of  a  bar-magnet  and  the  deflection  of  a  mag- 
netic needle  produced  by  this  same  bar-magnet  when  placed  at 
known  distances  E  and  W  (magnetically)  of  the  needle.  The 
dimensions  and  mass  of  the  magnet  must  also  be  obtained  in 
order  that  its  moment  of  inertia  may  be  calculated. 

If  the  period  of  vibration  of  the  magnet  be  T  in  the  place  in 


HORIZONTAL   COMPONENT   OF   EARTH  S    MAGNETIC   FIELD.    155 

which  we  wish  to  determine  the  horizontal  component  H,  its 
magnetic  moment  be  M,  and  its  moment  of  inertia  /,  then 


For,  when  the  magnet  is  deflected  through  a  small  angle  6,  the  restoring 
couple  is  MPI  sin  6  =  MH6.  Hence  if  the  angular  acceleration  at  that  moment 
is  a 

-MHd  =  Ia 
and 

_MH 
I 

Since  M,  H,  and  /  are  constant,  the  motion  is  simple  harmonic  and  T  is  given 

by  (i). 

If  a  magnetic  meedle  at  a  distance  d,  E  or  W  of  this  same  bar- 
magnet,  in  line  with  it  and  the  point  where  H  is  to  be  determined, 
be  deflected  through  an  angle  <f>  and,  when  at  a  distance  d\,  be 
deflected  through  an  angle  $1 

/  N  M    d5  tan  <£  —  d,  tan  <£i 

(2)     .        .        .       -7=  =  - 


Equation  (2)  is  deduced  from  the  expression  for  the  force  F  produced  by 
a  magnet  of  magnetic  moment  M  at  a  distance  d  in  the  direction  of  the  axis 
of  the  magnet.  For,  if  ra  is  the  strength  of  either  pole  of  the  magnet  and  2/ 
its  magnetic  length,  the  resultant  force  due  to  the  two  poles  is 


.  2Md 

r  — 


-iy   (d+iy   (d2-/2)2   (<*2-/2)2 

By  expanding  the  denominator  we  may  also  write  this: 


3«v"fiP/' 


in  which  K  is  approximately  a  constant.  (If  the  length  of  the  needle  were  also 
taken  account  of,  this  expression  would  remain  unchanged,  except  that  the 
value  of  the  constant  K  would  be  different).  If,  under  the  force  F  and  the 
component  H  of  the  earth's  magnetic  field,  a  magnetic  needle  makes  an  angle 
<f>  with  the  magnetic  meridian,  F  =  H  tan  <f>.  Hence, 

d3  tan  <J> 


If,  now,  the  distance  be  changed  to  di,  and  the  deflection  becomes  fa.  an- 
other equation  similar  to  the  above  will  be  obtained  and  the  elimination  of  K 
will  give  equation  (2)  above. 

From  equations  (i)  and  (2)  both  H  and  M  may  be  obtained 
when  the  other  quantities  have  been  measured. 


ELECTRICITY   AND   MAGNETISM. 

(A)  To  determine  the  period  of  vibration,  remove  all  movable 
iron   (knives,  keys,  etc.,  included)   to  several  meters  from  the 
vicinity  of  the  entire  experiment.     Suspend  the  deflecting  mag- 
net, by  means  of  a  stirrup  attached  to  a  single  strand  of  silk 
thread,  in  a  box  which  has  glass  ends  and  sides  and  is  surmounted 
by  a  glass  tube  through  which  the  suspension   passes.    Level 
until  the  thread  hangs  in  the  axis  of  the  tube.    The  magnet  may 
be  adjusted  until  it  is  horizontal  as  tested  by  comparison  with 
a  leveled  rod  attached  to  the  outside  of  the  box,  but  a  simple 
calculation  will  show  that,  if  the  error  in  the  adjustment  does 
not  exceed  5°,  the  error  in  the  period   does   not   exceed  i%. 
Attach  pointers  to  the  opposite  glass  sides  of  the  box  (or  adjust 
those   provided)  so  that  they  are  in  line  with  the  magnet  at  rest. 
Set  the  magnet  vibrating  through  an  angle  not  exceeding  10°. 
Check  any  pendulum  vibrations  by  judiciously  pressing  on  the 
top  of  the  glass  tube.    Then  determine  the  period  by  the  method 
of  passages  as  in  Exp.  X  (see  p.  49). 

The  magnet  should  be  vibrated  as  near  as  is  convenient  to  the 
place  where  the  needle  is  deflected  in  the  second  part,  i.e.,  where 
we  wish  to  determine  H. 

(B)  The  instrument  used  in  the  deflection  part  of  the  experiment 
is  called  a  magnetometer.    It  consists  of  a  box  with  glass  sides  in 
which  is  suspended  a  mirror  attached  to  either  a  small  magnetic 
needle  with  a  damping  vane  or  a  small  bell  magnet  vibrating  in 
a  copper  sphere.    The  sphere  is  placed  at  the  center  of  a  gradu- 
ated bar  upon  which  can  be  placed  the  deflecting  magnet.    Level 
until  the  suspending  fiber  is  at  the  center  of  the  bottom  of  the 
suspension-tube.     If  the  needle  or  the  damping  vane  does  not 
swing  free,  a  little  additional  leveling  will  be  necessary. 

A  specially  mounted  large  compass  needle  is  used  to  adjust  the 
magnetometer  bar  perpendicular  to  the  magnet  meridian.  By 
means  of  it  a  rod  is  placed  in  the  direction  of  the  magnetic  mer- 
idian, and  then,  by  means  of  a  square,  the  magnetometer  bar  is 
made  perpendicular  to  the  rod.  Place  a  telescope  and  scale 
about  one  meter  from  the  magnetometer.  See  that  the  scale  is 
perpendicular  to  the  telescope.  Adjust  until  the  scale  reflected 
from  the  mirror  is  clearly  seen  in  the  telescope  (for  directions  for 
this  adjustment  see  p.  23). 


HORIZONTAL  COMPONENT  OF  EARTH  S  MAGNETIC  FIELD.      157 

Place  the  magnet  whose  period  of  vibration  has  been  deter- 
mined on  a  small  wood  slide  near  one  end  of  the  magnetometer 
bar.  Note  the  scale-reading  on  the  magnetometer  bar  corre- 
sponding to  the  end  of  the  magnet  nearer  the  needle.  When  the 
needle  comes  to  rest,  record  the  scale-reading  against  the  vertical 
cross-hair  of  the  telescope.  Remove  the  magnet  several  meters 
and  read  the  zero.  Replace  the  magnet  at  the  same  distance  from 
the  needle,  but  reversed,  and  again  read  the  scale  division  cor- 
responding to  the  vertical  cross-hair.  Make  two  similar  read- 
ings with  the  magnet  at  an  equal  distance  on  the  other  side  of 
the  needle.  Read  the  zero  before  or  after  each  reading  and  always 
estimate  tenths  of  millimeters.  Make  four  similar  readings  with  the 
magnet  at  about  two-thirds  the  distance  on  each  side  of  the  needle. 

If  the  zero  is  somewhat  unsteady,  the  following  method  will 
be  found  better.  Omit  zero  readings  and  obtain  the  four  deflec- 
tion readings  as  rapidly  as  possible.  Do  this  three  times  for 
each  distance  so  that  twelve  readings  for  each  distance  are  attained. 
Take  half  the  difference  of  each  two  successive  readings  as  one 
value  of  the  deflection.  The  final  result  will  be  the  mean  of  all 
values  so  found.  The  extent  to  which  they  agree  will  indicate 
the  reliability  of  the  mean. 

Measure  the  distance  from  the  center  of  the  scale  beneath  the 
telescope  to  the  center  of  the  suspension-tube  of  the  magneto- 
meter (i.  e.,  the  distance  to  the  mirror).  From  this  distance  and 
the  mean  scale-reading  for  that  distance,  tan  20  is  obtained 
(for  it  must  be  remembered  that  a  reflected  ray  of  light  is  turned 
through  twice  the  angle  that  the  reflecting  mirror  is  turned 
through)  .  Since  0  is  a  small  angle  tan  20  =  2  tan  0  very  nearly. 
The  distances  from  the  needle  to  the  near  end  of  the  magnet  plus 
half  the  length  of  the  magnet  give  d  and  di.  At  the  close  of  the 
experiment,  measure  the  length  of  the  magnet  with  vernier  cal- 
ipers, and  the  diameter  with  micrometer  calipers,  and  also  weigh 
it.  If  /  be  the  length,  r  the  radius,  and  m  the  mass,  the  moment 
of  inertia  is: 


In  reporting,  state  the  possible  errors  of  the  measurements  of 
,  7,  d,  di,  tan  0,  tan  0i. 


158  ELECTRICITY   AND   MAGNETISM. 

Questions. 

1.  How  could  the  true  length  of  the  deflecting  magnet  be  obtained? 

2.  H  having  been  obtained  at  one  point  in  the  room  or  building,  what  would 
be  the  easiest  way  of  finding  its  value  at  any  other  point? 

3.  What  are  the  other  "elements"  of  the  earth's  magnetism? 

4.  If  you  have  done  Exp.  XLIII,  calculate  the  total  force  and  the  vertical 
component. 

5.  State  reasons  for  placing  (a)  magnetometer  needle  in  (B)  where  magnet 
was  vibrated  in  (A);  (b)  magnetometer  bar  East  and  West;  (c)  telescope  scale 
parallel  to  mirror. 


XLIII.    MAGNETIC  INCLINATION  OR  DIP. 

(A)  Dip  Circle. 

References — Elementary:  Duff,  §§386-387;  Ames,  pp.  618-619;  Crew,  §297; 
Hadley,  pp.  99-102;  Kimball,  §494;  Reed  &  Guthe,  §249;  Spinney,  §306; 
Watson,  §§429-431. — More  Advanced:  Stewart  &  Gee's  Practical  Physics, 
II,  pp.  275-284. 

The  dip,  or  inclination  of  the  earth's  magnetic  lines  of  force 
to  the  horizontal,  is  found  by  means  of  a  dipping  needle  or  mag- 
netic needle  suspended  on  a  horizontal  axis  which  passes  as  nearly 
as  possible  through  the  center  of  gravity  of  the  needle,  with  a 
vertical  graduated  circle  for  reading  the  angle  of  inclination. 
Such  an  apparatus  is  called  a  dip  circle,  and  includes  a  level  and 
leveling  screws  for  making  the  circle  vertical,  knife-edges  for 
bearing  the  axis  of  the  needle,  a  horizontal  graduated  circle  for 
fixing  the  azimuth  of  the  vertical  circle,  and  an  arrestment,  with 
Y-shaped  supports,  for  raising  and  lowering  the  needle  and  placing 
it  so  that  its  axis  of  rotation  passes  as  nearly  as  possible  through 
the  center  of  the  vertical  circle. 

The  zero-line  of  the  vertical  circle  must  first  be  made  vertical. 
This  adjustment  is  made  by  means  of  the  leveling  screws  and 
level  just  as  a  cathetometer  is  leveled  (see  p.  18).  The  circle  must 
then  be  turned  into  the  plane  of  the  magnetic  meridian.  To 
attain  this,  advantage  is  taken  of  the  fact  that  if  the  plane  in 
which  the  needle  is  free  to  rotate  be  at  right  angles  to  the 
magnetic  meridian,  the  needle  must  stand  vertically;  for  in  that 
position  the  horizontal  component  of  the  earth's  magnetic  force 
is  parallel  to  the  axis  of  rotation  of  the  needle,  and  hence  has  no 
moment  about  that  axis.  The  circle  is,  therefore,  turned  approxi- 
mately east  and  west  and  then  adjusted  until  the  needle  is  ver- 


MAGNETIC    INCLINATION   OR   DIP.  159 

tical.  This  adjustment  should  be  repeated  several  times,  and 
each  position  should  be  carefully  read  with  the  assistance  of  a 
vernier  if  one  is  provided.  *  A  rotation  of  the  circle  through  90° 
from  the  mean  position,  as  indicated  by  the  horizontal  circle, 
should  then  bring  the  plane  of  the  circle  to  coincidence  with  the 
plane  of  the  magnetic  meridian.  By  raising  and  lowering  the 
arrestment,  the  needle  is  then  placed  on  the  knife-edges  in  the 
proper  position  for  indicating  the  dip. 

A  single  reading  of  the  needle  in  this  position  would  give  a 
very  imperfect  value  of  the  dip.  Errors  arise  from  various  causes : 
(i)  the  axis  may  not  roll  freely  on  the  knife-edges,  owing  to 
dust  or  friction.  To  remove  any  dust  the  axis  and  knife-edges 
should  be  brushed  with  a  camel's  hair  brush.  The  setting  by 
means  of  the  arrestment  and  the  readings  should  be  made  at  least 
twice,  and  both  sets  of  readings  recorded.  (2)  The  axis  of  rota- 
tion of  the  needle  may  not  be  exactly  at  the  center  of  the  divided 
circle.  This  error  may  be  eliminated  by  reading  the  position  of 
both  ends  of  the  needle,  one  reading  being  from  this  cause  as 
much  too  great  as  the  other  is  too  small.  (3)  The  line  of  zeros 
on  the  vertical  scale  may  not  be  truly  vertical,  and  this  would 
cause  errors  in  the  same  direction  in  the  readings  of  the  ends  of 
the  needle.  These  errors  may  be  eliminated  by  turning  the  ver- 
tical circle  through  180°  about  a  vertical  axis  and  repeating  the 
readings,  for  in  these  readings  the  quadrants  on  the  other  side 
of  the  zero  line  are  used.  (4)  The  axis  of  rotation  may  not  pass 
exactly  through  the  center  of  gravity  of  the  needle.  So  far  as 
the  fault  lies  in  the  fact  that  the  axis  of  rotation  is  to  one  side  of 
the  axis  of  figure  of  the  needle,  the  error  may  be  eliminated  by 
reversing  the  needle  in  its  bearings  and  repeating  the  readings; 
for  in  one  position  gravity  will  make  the  readings  as  much  too 
great  as  in  the  other  case  it  makes  them  too  small.  But  gravity 
will  also  cause  an  error  if  the  axis  of  rotation  be  in  the  axis  of 
figure,  but  not  at  the  center  of  the  latter.  The  error  will  not  be 
eliminated  by  reversing  the  needle  on  its  bearings,  but  it  will  be 
if  the  magnetism  of  the  needle  is  reversed  and  all  of  the  preced- 
ing readings  repeated;  for  then  the  other  end  of  the  needle  will 
be  lower  and  the  error  will  be  in  the  opposite  direction.  The  re- 


160  ELECTRICITY   AND   MAGNETISM. 

versal  of  the  magnetism  should  be  done  under  the  direction  of 
the  instructor,  the  method  of  double  touch  being  used. 

In  recording  these  various  positions  and  readings,  the  side  of 
the  circle  on  which  the  scale  is  engraved  may  be  called  the  face 
of  the  instrument,  and  similarly  one  side  of  the  needle  may  be 
fixed  upon  as  its  face.  Thus  two  readings  of  each  end  of  the 
needle  are  to  be  made  in  each  of  the  following  positions : 

(1)  Face  of  instrument  E,  face  of  needle  E; 

(2)  Face  of  instrument  W,  face  of  needle  W; 

(3)  Face  of  instrument  W,  face  of  needle  E; 

(4)  Face  of  instrument  E,  face  of  needle  W. 

The  magnetism  of  the  needle  having  been  reversed,  readings 
are  to  be  again  taken  in  the  above  positions.  The  final  result  is 
taken  as  the  mean  of  these  32  readings. 

(B)  Earth  Inductor. 

References — Elementary:  Duff,  §515;  Hadley,  p.  425;  Kimball,  §714;  Wats.n, 
§520. — More  Advanced:  Ayrton  &  Mather's  Practical  Electricity,  §§144-149; 
Watson  (Pr.),  §§218,  220,  223. 

Another  instructive  method  of  determining  the  dip  is  by  means 
of  an  earth  inductor  in  series  with  a  ballistic  galvanometer 

(P-  I47-) 

When  the  earth  inductor  is  placed  perpendicular  to  a  magnetic 
field  and  then  reversed,  the  flow  of  electricity  due  to  induction 
in  the  circuit  is  proportional  to  the  magnetic  flux  and  hence  to 
the  strength  of  the  field  perpendicular  to  the  coil  (see  text-book 
references  on  earth  inductor  and  electromagnetic  induction). 
Since  the  reversal  is  not  instantaneous,  either  a  ballistic  galvan- 
ometer of  very  long  period  or  a  flux  galvanometer  (p.  148) 
should  be  used. 

The  earth  inductor  is  first  placed  with  the  plane  of  its  coils 
vertical  and  perpendicular  to  the  magnetic  meridian.  It  is  then 
rotated  through  1 80°  and  the  throw  di  noted.  Several  readings 
should  be  made.  The  plane  of  the  coils  is  then  placed  horizon- 
tally and  the  throw  d%  on  rotation  through  180°  noted.  The 
ratio  of  dz  to  d\  is  the  tangent  of  the  dip. 


MEASUREMENT   OF    RESISTANCE.  l6l 

Questions. 

1.  State  the  error  eliminated  in  each  set  of  readings  with  dip  circle. 

2.  What  other  sources  of  error  may  there  be  in  measurement  by  the  dip 
circle? 

3.  Would  you  be  justified  in  making  a  calculation  of  "probable  error"  from 
the  various  readings  with  the  dip  circle? 

4.  If  you  have  performed  Exp.  XLII,  calculate  the  total  force  and  the 
vertical  component. 

5.  How  could  the  dipping  needle  be  used  to  compare  the  vertical  intensities 
as  two  different  points  ? 

6.  What  additional  information  is  required    to   determine    the    intensity 
from  (B)? 


XLIV.    MEASUREMENT  OF  RESISTANCE  BY  WHEAT- 
STONE'S  BRIDGE. 

References — Elementary:  Duff,  §§444-447,  456;  Ames,  pp.  725-727;  Crew,  §382; 
Hadley,  pp.  306-309;  Kimball,  §§646-652;  Reed  &  Guthe,  §§272-274; 
Spinney,  §§284,  288;  Watson,  §§481,  488. — More  Advanced:  Ayrton  & 
Mather's  Practical  Electricity,  §§88-98;  Kohlrausch's  Physical  Measure- 
ments, p.  303;  Watson  (Pr.),  §§176-177. 

The  practical  measurement  of  a  resistance  consists  in  comparing 
it  with  a  known  or  standard  resistance.  For  resistances  of  me- 
dium magnitude,  Wheatstone's  Bridge  is  usually  used  (p.  144). 

In  joining  the  known  and  unknown  resistances  to  the  bridge, 
connectors  should  be  used  whose  resistance  is  negligible ;  that  is, 
less  than  the  unavoidable  error  that  may  occur  in  determining 
the  unknown  resistance.  In  connecting  the  battery  and  galvan- 
ometer, no  such  precaution  is  necessary,  for  their  resistances  do 
not  enter  int6  the  calculation.  The  galvanometer  may  be  con- 
nected to  either  pair  of  opposite  corners;  but,  where  the  great- 
est sensitiveness  is  required,  if  the  galvanometer  has  a  higher 
resistance  than  the  battery,  it  should  be  in  the  branch  that  con- 
nects the  junction  of  the  highest  two  of  the  four  resistances  P, 
Q,  R,  S  to  the  junction  of  the  lowest  two;  while,  if  the  battery 
has  the  greatest  resistance,  it  should  occupy  that  position.  Two 
spring  keys  should  be  included  in  the  connections,  one  in  the 
battery  arm  and  the  other  in  the  galvanometer  arm.  When  test- 
ing for  a  balance,  the  battery  key  should  be  pressed  first,  then 
the  galvanometer  key.  If  taken  in  the  reverse  order,  there 
might  be  a  small  deflection  due  to  the  self-induction  of  the  vari- 
ous parts.  These  keys  should  be  pressed  for  a  moment  only, 
ii 


1 62  ELECTRICITY   AND   MAGNETISM. 

Except  for  a  final  determination,  it  is  not  necessary  to  wait 
until  the  galvanometer  has  quite  come  to  rest,  for  a  lack  of  bal- 
ance will  be  indicated  by  a  sudden  disturbance  of  the  swing  when 
the  galvanometer  key  is  pressed.  The  pressure  of  the  galvano- 
meter key  should  be  brief,  sufficient  merely  to  indicate  the  direc- 
tion of  the  initial  movement. 

Do  not  at  first  seek  zero  deflection  of  the  galvanometer  but  ad- 
just known  resistance  and  position  of  sliding  contact  until  a 
slight  change  in  either  produces  a  reversal  in  the  direction  of  the 
galvanometer  deflection.  Then,  if  possible,  refine  the  adjustment 
until  the  deflection  is  zero. 

In  practice,  it  is  best  to  use  a  box-resistance  as  nearly  as  pos- 
sible equal  to  the  unknown  resistance.  This  comes  to  the  same 
thing  as  saying  that  the  box-resistance  should  be  varied  until  a 
balance  is  attained  when  the  parts  of  the  meter  wire  are  nearly 
equal.  The  reason  for  this  preference  is  that  the  sensitiveness 
is  then  a  maximum,  or  a  slight  lack  of  balance  is  most  easily 
detected  by  the  deflection  of  the  galvanometer.  The  exact  ratio 
of  P  to  Q  for  a  balance  should  be  very  carefully  ascertained. 
At  least  six  settings  should  be  made ;  and  to  secure  independence 
of  the  settings,  the  eye  should  be  kept  on  the  galvanometer-scale 
and  the  reading  of  the  bridge  not  examined  until  the  setting  has 
been  decided  on.  The  mean  of  these  six  is  then  taken.  R  and 
S  should  then  be  interchanged  and  six  more  settings  made.  This 
interchange  will  serve  to  eliminate  the  effect  of  lack  of  symmetry 
of  the  two  sides  of  the  wire  bridge -and  its  connections. 

The  structure  of  the  galvanometer  to  be  used,  its  coils,  magnets, 
and  connections,  should  be  carefully  examined  and  care  taken 
that  it  is  thoroughly  understood  (p.  145). 

Three  unknown  resistances  should  be  measured  separately  and 
then  all  in  parallel.  From  the  separate  resistances  the  resist- 
ance of  the  conductors  in  parallel  should  be  calculated  and  com- 
pared with  the  measurement  of  the  same.  The  resistance  of  a 
wire  should  then  be  measured  and  its  length  and  mean  diameter 
obtained.  From  these  data,  the  specific  resistance  of  the  material 
of  the  wire  should  be  deduced.  The  temperature  at  which  the 
resistance  is  measured  should  also  be  noted,  and  from  the  tern- 


GALVANOMETER    RESISTANCE   BY   SHUNT   METHOD.  163 

perature  coefficient  of  the  material   (Table  XXII)  the  specific 
resistance  at  o°  C.  calculated. 

The  possible  errors  of  the  measurements,  and  hence  the  extent 
to  which  the  calculations  should  be  carried,  may  be  deduced  from 
the  mean  deviation  in  each  set  of  readings. 


Questions. 

1.  Does  the  battery  need  to  be  a  constant  one? 

2.  What  objections  are  there  to  allowing  the  battery  circuit  to  remain 
closed? 

3.  Why  is  it  difficult  by  this  method  to  measure  very  large  or  very  small 
resistances? 


XLV.    GALVANOMETER  RESISTANCE  BY  SHUNT 
METHOD. 

References  —  Ayr  ton  &  Mather's  Practical  Electricity,  §109;  Kohlrausch's  Physical 
Measurements,  p.  325. 

If  a  galvanometer  of  resistance  G  connected  in  series  with  a 
battery  of  resistance  B  and  e.  m.  f.  E  and  a  box  resistance  R 
gives  a  deflection  d  and  if  C  be  the  current 

r-        E        -Kd 
~R+B+G~ 

where  K  is  a  constant  for  the  galvanometer.  If  now  the  galvano- 
meter be  shunted  by  a  resistance  S  and  the  deflection  be  then  df 
and  the  current  through  the  galvanometer  C', 

'  E  —       ' 

GS 


G+S 

Hence 

(R+B)(G+S)+GS=d 
S(R+B+G)  d" 

and  from  this  G  is  readily  deduced  provided  B  is  kmown. 
Usually  a  battery  of  such  low  resistance  can  be  used  that  B  is 
negligible  compared  with  R  and  may  be  omitted;  otherwise  B 
must  be  obtained  as  in  Exp.  LIII.  The  galvanometer  should  be 


164 


ELECTRICITY   AND   MAGNETISM. 


connected  through  a  commutator  and  several  readings  on  both 

sides  should  be  made., 

If  the  e.  m.  f.  of  the  cell  supplied  is  too  great,  a  suitable  frac- 
A  A    -.  tion    of    it    should    be  employed;    note  pre- 

caution as  to  magnitude  of  resistances  (see 
§35 »  P-  I53)-  As  a  check,  the  determination 
of  G  should  be  repeated,  a  different  value  for 
S  being  used.  If  the  galvanometer  is  very 
sensitive,  its  resistance  must  be  found  from 
two  readings  with  shunts.  A  suitable  formula 
is  readily  worked  out. 

If  R  should  be  very  great  compared  with 
the  other  resistances,  the  formula  may  be  sim- 
plified. This  will  usually  be  the  case  if  the 
galvanometer  is  very  sensitive  or  of  low  re- 
sistance. The  quantities  added  to  R  in  the 


FIG.  51. 
first  two  equations  may  then  be  neglected  and  we  get 


G+S=d 
S       dr 

Care  must,  however,  be  taken  to  ascertain  that  the  above  con- 
ditions are  sufficiently  closely  satisfied.  This  is  assured  if  very 
different  values  of  5  give  the  same  value  for  G,  or  it  can  be  ascer- 
tained by  calculation  from  approximate  values  of  G  and  B." 


XLVI.     GALVANOMETER    RESISTANCE    BY    THOMSON'S 

METHOD. 

References — Kohlrausch's  Physical  Measurements,  p.  328;  Stewart  &  Gee's  Prac- 
tical Physics,  II,  p.  140-142. 

The  resistance  of  the  coils  of  a  galvanometer  may  be  found  by 
means  of  Wheatstone's  Bridge  as  the  resistance  of  any  ordinary 
conductor  is  found.  This  would  require  the  use  of  a  second  gal- 
vanometer. The  second  galvanometer,  for  detecting  when  the 
bridge  is  balanced,  is  frequently  unnecessary.  The  condition  for 
a  balance  is  that,  when  the  branch  in  which  the  galvanometer  is 
usually  placed  is  closed  by  a  key,  no  current  shall  flow  through 


GALVANOMETER  RESISTANCE  BY  THOMSON'S  METHOD.         165 


it.  If  a  current  did  flow  through  it,  a  change  would  take  place 
in  the  currents  in  the  other  arms.  Now  the  presence  of  a  galvan- 
ometer in  one  of  these  arms  enables  us  to  test  whether  any  change 
in  the  distribution  of  the  currents  takes  place  on  the  key's  being 
pressed.  Hence,  in  Thomson's  method  for  galvanometer  resist- 
ance the  galvanometer  is  placed  in  the  "unknown"  arm  and  a 
spring  key,  K,  is  placed  in  the  branch  in  which,  in  the  ordinary 
arrangement  of  Wheatstones'  Bridge,  a  galvanometer  is  found. 
A  diagram  to  illustrate  the  connections  is  given  in  Fig.  52. 

From  the  above  it  will  be  seen  that  in  this  method  a  balance  is 
obtained  when  the  deflection  of  the  galvanometer  does  not  change  on 
the  key,  K,  being  pressed.  Two 
practical  difficulties  are  met  with. 
The  first  is  that  the  deflection  of 
the  .galvanometer  before  the  key 
is  pressed  may  be  so  large  that 
it  cannot  be  read.  When  the 
galvanometer  is  of  the  Thomson 
type  (p.  146),  this  difficulty  may 
be  overcome  by  turning  the  con- 
trol magnet  until  the  deflection 
can  be  read  (the  zero  position 
of  the  galvanometer  could,  of 
course,  not  then  be  read  on  the 
scale,  but  that  is  not  necessary). 
In  the  d'Arsonval  type  of  galvanometer  there  is  no  such  way  of 
overcoming  this  difficulty,  and  so  this  method  is  not  so  easily 
applied  to  such  a  galvanometer.  The  second  difficulty  is  that 
if  the  battery  be  a  variable  one,  the  galvanometer  will  not  give 
a  steady  deflection.  Hence,  a  constant  battery  of  the  Daniell 
or  Gravity  type  should  be  used  (p.  152).  It  may  also  be  neces- 
sary to  decrease  the  current  through  the  bridge  and  galvanometer 
by  putting  considerable  resistance  in  series  with  the  battery,  or 
a  fraction  of  the  e.  m.  f.  of  the  cell  may  be  used  (p.  153). 

Polarization  of  the  cell  will  probably  cause  a  somewhat  slow 
drift  of  the  galvanometer  even  when  a  true  balance  may  exist. 
For  this  reason  the  key  K  should  be  pressed  only  for  a  •moment 


FIG.  52. 


1 66  ELECTRICITY   AND   MAGNETISM. 

and  the  closest  limits  giving  opposite  deflections  should  be  as- 
certained. 

In  the  experiment  it  is  better  to  use  a  bridge-box  instead  of  a 
wire  bridge,  for  the  condition  for  sensitiveness,  that  the  arms 
should  be  as  nearly  equal  as  possible,  still  holds,  and  the  resist- 
ance of  a  wire  bridge  is  usually  very  small  compared  with  that 
of  the  galvanometer.  Beginners  sometimes  find  difficulty  in 
deciding  on  the  proper  connections.  The  best  way  is  to  consider 
what  the  connections  would  be  in  the  ordinary  use  of  Wheat- 
stone's  Bridge,  and  then  consider  the  modifications  introduced 
in  the  present  method.  If  possible,  ratio  arms  of  1000  to  1000, 
100  to  1000,  and  10  to  1000  should  be  used  in  succession  to  ob- 
tain successive  approximations.  The  last  should  give  the  resist- 
ance to  two  places  of  decimals  (if  one  ohm  is  the  least  box-resist- 
ance), but  the  decreasing  sensitiveness  may  prevent  the  latter 
ratios  from  giving  more  accurate  results  than  the  first. 

If  the  galvanometer  has  more  than  one  coil,  the  resistance  of 
each  should  be  measured  separately  and  then  the  resistance  of 
all  in  series.  This  wilt  afford  a  check  on  the  work. 


Questions. 

1.  Describe  carefully  the  essential  parts  of  the  galvanometer  used.     State 
its  type  and  class. 

2.  What  is  meant  by  the  polarization  of  a  cell  and  to  what  is  it  due? 


XLVn.    MEASUREMENT  OF  HIGH  RESISTANCES  (i). 

The  method  of  Wheatstone's  Bridge  is  not  suitable  for  measur- 
ing very  high  resistances.  One  method  is  to  connect  the  unknown 
resistance  X,  a  battery  of  negligible  resistance  and  e.  m.  f.  E, 
and  a  sensitive  galvanometer  of  resistance  G  in  series.  If  the 
current  be  C, 

p< 
C  —         ^,  giving  a  deflection  d. 

J\.  ~T"Cj 

Now  replace  X  by  a  known  resistance,  R,  and  shunt  the  galvan- 
ometer by  such  a  resistance,  5,  that  the  deflection  is  readable. 
By  considering  the  total  current  and  the  part  C'  of  the  total 


MEASUREMENT   OF   HIGH   RESISTANCES.  167 

current  that  passes  through  the  galvanometer,  we  readily  find 
that 

T?  S* 

C'  =  —       „„    „  ,  0>  giving  a  deflection  d' '. 

p    ,        &:>      CrH-O 

R+G+S 

Hence, 

R(G+S)+GS=d 
S(X+G)         dn 

and  from  this  X  is  readily  deduced.  G  may  be  found  as  in  Exp. 
XLV  or  XLVI;  but  if  (G  +  S)  /S  is  known  and  G  is  small  com- 
pared with  X  and  R,  the  resistance  of  the  galvanometer  need 
not  be  determined.  Many  galvanometers  are  provided  with 
shunt  boxes,  for  which  S/(G  +  S)  is  o.r,  o.oi,  or  o.ooi. 

The  galvanometer  should  be  connected  through  a  commutator, 
and  several  readings  on  both  sides  should  be  made  to  obtain  a 
reliable  mean. 

As  a  check,  repeat  the  measurements  with  a  different  value  for 
R  and  a  different  value  for  5. 


XLVIII.    MEASUREMENT  OF  HIGH  RESISTANCES  (2). 

References — Elementary:  Duff,  §407;  Ames,  pp.  658-659;  Hadley,  p.  208; 
Kimball,  §559;  Watson,  §467. — More  Advanced:  Ayrton  &  Mather's  Prac- 
tical Electricity,  §49;  Henderson's  Electricity  and  Magnetism,  pp.  71-75; 
Watson  (Pr.),  §241. 

A  very  high  resistance,  such  as  the  insulation  resistance  of  a 
cable  01  the  resistance  of  cloth,  paper,  wood,  etc.,  may  be  measured 
by  finding  the  rate  at  which  the  electricity  in  a  charged  con- 
denser leaks  through  the  conductor.  An  electrometer  is  used  to 
find  the  change  of  potential  of  the  condenser  and  from  this  the  rate 
of  loss  of  its  charge  is  deduced.  The  Dolezalek  form  of  Kelvin's 
quadrant  electrometer  is  suitable.  Its  needle  is  kept  charged  to 
a  high  potential  by  being  connected  to  one  pole  of  a  battery  of 
small  cells,  the  other  pole  being  grounded. 

To  find  the  insulation  resistance  of  a  cable  the  whole  of  the 
cable  except  the  ends  is  immersed  in  a  tank  of  salt  water  which  is 
connected  to  the  earth.  One  of  the  ends  is  carefully  paraffined 


1 68 


ELECTRICITY   AND   MAGNETISM. 


to  prevent  surface  leakage  and  the  core  of  the  other  end  is  con- 
nected to  the  insulated  pair  of  quadrants.  If  the  cable  is  sheathed 
with  metal,  immersion  is  not  necessary.  Other  materials,  such 
as  those  mentioned,  are  pressed  between  sheets  of  tinfoil,  one 
sheet  being  connected  to  -the  earthed  quadrants  and  the  other 
to  the  insulated  quadrants. 


FIG.  53- 

Let  Vi  =  potential  given  the  condenser  on  closing  the  key  K. 
The  charge  Q  in  the  condenser  and  cable  =  C  Vi  where  C  is  their 
joint  capacity.  Upon  opening  the  key  the  charge  flows  through  a 
resistance  RI,  for  a  time,  /,  RI  being  the  insulation  resistance 
of  the  cable,  the  condenser,  and  the  electrometer  and  keys  in 
parallel. 

Since  the  current  at  the  time  /  equals  V/R\  by  Ohm's  law,  arid 
also  equals  the  rate  of  decrease  of  Q  or  of  CV 


_  =  _r 


dV 


dt 
dV     dt 


Integrating  between  the  limits  t—o  when  V=  V\  and  /  =  /  when 
V  =  V%  we  get 

Vi      t 


0-434 


where  d\  and  d%  are  the  initial  and  final  deflections  of  the  electro 
meter  from  the  zero  position. 


MEASUREMENT   OF   LOW    RESISTANCES.  169 

The  zero  should  be  determined  both  before  V\  is  found  and 
after  V2  is  found.  As  it  is  very  apt  to  vary  slightly,  more  reliable 
results  can  be  attained  by  continuing  to  read  V  at  intervals 
(e.  g.,  every  half-minute)  until  it  has  fallen  to  about  one-half 
of  its  original  value.  From  a  curve  drawn  to  represent  V  and  /, 
two  reliable  points  may  be  chosen  to  give  values  for  V\  and  V2  to 
be  used  in  the  calculation. 

A  subdivided  condenser  is  desirable  in  order  that  a  capacity 
giving  a  sufficiently  rapid  fall  of  potential  may  be  chosen. 

The  total  insulation  resistance,  R2,  of  the  other  parts  in  par- 
allel with  the  cable  are  found  by  disconnecting  the  cable  and  mak- 
ing a  second  set  of  observations  as  above.  The  insulation  re- 
sistance, R,  of  the  cable  is  then  deducible,  for 

JL  =  1_4_.L 
R!     R  ^Rz 

The  capacity,  C\,  of  the  cable  can  be  compared  with  that  of  the 
condenser,  £2,  by  the  method  of  "divided  charge."  First  charge 
the  condenser  and  observe  its  potential  by  the  electrometer  and 
let  the  deflection  be  d\.  Then  connect  in  the  cable  and  let  d2  be 
the  new  deflection.  Since  the  total  charge  Q  remains  unchanged, 


and,  since  the  deflections  are  proportional  to  the  potentials, 
C—  C\-\-  C<L  =  ^Ci     and    C\  =  Ci(d\  —  dz)/dz. 


Questions. 

1.  Why  should  one  pole  of  the  battery  that  charges  the  needle  be  grounded? 

2.  Why  must  keys  of  specially  high  insulation  be  used  in  this  method? 

3.  Calculate  the  capacity  of  the  cable  in  electrostatic  units  from  rough 
measurements  of  its  dimensions  and  reduce  to  microfarads  (see  p.  154). 

XLIX.    MEASUREMENT  OF  LOW  RESISTANCES  (i). 

References  —  Henderson's  Electricity  and  Magnetism,  pp.  57-58;  Stewart  &  Gee's 
Practical  Physics,  II,  pp.  177-181;  Watson  (Pr.),  §§190-194. 

Very  low  resistances  cannot  be  measured  by  the  Wheatstone 
Bridge  method,  because  the  unknown  resistances  of  the  connec- 
tions are  not  small  compared  with  the  resistance  to  be  measured. 
The  simplest  method  for  low  resistances  is  a  "fall  of  potential" 
method.  A  current  is  passed  through  the  resistance,  the  current 


170 


ELECTRICITY   AND   MAGNETISM. 


is  measured  by  an  ammeter  and  the  difference  of  potential  is 
measured  by  a  voltmeter;  then  the  resistance  is  known  from 
Ohm's  Law.  For  very  low  resistances  the  fall  of  potential 
will  be  very  small  and  an  instrument  much  more  sensitive  than 
any  commercial  voltmeter  must  be  used.  Instead  of  a  voltmeter 
a  sensitive  galvanometer  of  high  resistance,  or  a  low  resistance 
galvanometer  in  series  with  a  high  resistance,  is  used  and  the 
value  of  a  scale  division  of  the  galvanometer  regarded  as  a  volt- 
meter is  found  by  a  separate  experiment. 

Let  the  resistance  to  be  measured  be  x,  and  let  the  difference  of 
potential  at  its  ends  when  current  C  passes  through  it  be  e.    Then 


• 


e 

*~c 


C,  which  should  be  large,  may  be  measured  by  an  ammeter. 
To  find  e  we  must  know  the  constant,  K,  of  the  galvanometer 
considered  as  a  voltmeter;  that  is,  the  number  of  volts  per  unit 
deflection.  If  the  deflection  is  D 


To  find  K  apply  to  the  galvanometer  a  small  fraction  of  the 
!  E  e.  m.  f.,  E,  of  a  Daniell's  cell  (p.   152). 

For  this  purpose  connect  the  cell  in  series 
with  a  very  high  resistance  box  and  a 
box  of  moderate  resistances  and  join  the 
galvanometer  to  the  ends  of  one  of  the 
small  resistances,  r,  choosing  r  so  that 
the  deflection,  d,  will  not  be  very  dif- 
ferent from  D.  Then  if  the  resistance  of 
the  galvanometer  be  great  compared  with 
r  (see  p.  153)  and  if  the  total  resistance 
FIG.  54.  m  series  with  the  battery  be  R,  the  e.  m.  f. 

acting  on  the  galvanometer  is  Er/R. 
Hence 

rE 


The  above  is  on  the  assumption  that  the  deflections  are  so 
small  as  to  be  proportional  to  the  currents  (p.  147).  In  any  case 
it  is  well  to  use  such  values  of  r  and  R  that  the  deflection  is 
approximately  the  same  as  before. 


MEASUREMENT   OF   LOW   RESISTANCES. 


171 


In  the  first  part  of  the  experiment  place  a  commutator  in  the 
main  circuit  so  that  C  may  be  reversed  and  the  effect  of  thermo- 
electric forces  at  the  contacts  eliminated,  and  connect  the  galvan- 
ometer through  a  second  commutator  so  that  lack  of  sym- 
metry in  its  deflection  may  be  eliminated,  by  always  taking  read 
ings  on  the  same  side. 

Exactly  similar  precautions  should  be  observed  in  the  second 
part.  Close  the  currents  only  for  the  shortest  possible  times 
necessary  to  make  the  readings,  otherwise  heating  may  occur 
and  resistances  (especially  the  unknown  x)  may  change. 

The  determination  should  be  repeated  several  times  with 
different  values  of  C.  If  the  work  has  been  reliable,  D  should  be 
proportional  to  C.  Note  also  the  temperature  of  the  specimen 
and  calculate  its  resistivity  from  its  resistance  and  dimensions. 

Questions. 

1.  If  r  had  not  been  negligible  compared  ^with  the  galvanometer  resistance 
how  would  this  have  appeared  in  the  course  of  the  work? 

2.  Find  the  equation  that  must  replace  the  above  if  the  resistance  of  the 
battery  is  not  negligible  compared  with  R  and  if  r  is  not  negligible  compared 
with  the  galvanometer  resistance. 

L.    MEASUREMENT  OF  LOW  RESISTANCES  (2). 

See  references  to  XLIX. 

When  a  standard  low  resistance  (o.oi  or  o.ooi  ohm)  is  available, 
a  conductor  of  low  resistance  x 
may  be  connected  in  series  with 
it  and  a  battery,  and  a  very  sensi- 
tive voltmeter,  or  a  high-resistance 
galvanometer,  serving  as  a  volt- 
meter, may  be  used  to  compare 
the  falls  of  potential  in  x  and  the 
standard.  The  resistances  will  be 
proportional  to  the  falls  of  poten- 
tial. 

Connection  with  the  battery 
should  be  made  through  a  com- 
mutator to  reverse  thermal  effects 
at  the  connections,  and  the  galvano- 
meter should  be  connected  through 
a  second  commutator  to  eliminate 


172 


ELECTRICITY   AND   MAGNETISM. 


asymmetry  of  the  galvanometer  readings   Thus  each  final  reading 
will  be  the  mean  of  four  separate  readings. 

The  currents  should  be  closed  for  the  shortest  times  sufficient 
for  the  readings,  to  avoid  heating.  Note  the  temperature  of  the 
specimen. 

Questions. 

i.  What  are  the  comparative  advantages  and  disadvantages  of  this  and 
the  preceding  method? 

3!  Will 
ance  by  Wheat-stone's  Bridge?     Why? 


Why  is  a  high-resistance  galvanometer  to  be  preferred? 
fill  poor  contact  have  as  much  effect  as  in  a  measurement  of  low  resist- 


LI.    MEASUREMENT  OF  LOW  RESISTANCES  BY  THE 
THOMSON  DOUBLE  BRIDGE. 

References— KoUrausch1  s  Text-book  of  Practical  Physics  (loth  Edition  German}, 
§93,  II  (a);  Stewart  &  Gee's  II,  pp.  182-187;  Watson  (Pr.),  §191. 

In  Thomson's  Double  Bridge  the  errors  of  the  contacts  in  the 
use  of  Wheatstone's  Bridge  are  avoided.  Its  principle  is,  in  fact, 
that  of  the  fall  of  potential  method 
(Exp.  L)  the  direct  comparison  of  the 
falls  of  potential  being  replaced  by  a 
null  method.  This  method  is  applicable 
to  extremely  low  resistances  as  well  as 
to  medium  resistances. 

In  the  diagram  x  is  the  resistance  to  be 
measured  and  r  a  standard  known  resist- 
ance; a  may  be  made  10  or  100;  and  b 
may  be  made  100,  1,000,  10,000.  Simi- 
larly, a'  may  be.  made  10  or  100  and  b' 
FIG  6  100,  1,000,  10,000.  Now  let  a,  a',  b,  b' 

be  taken  so  that  a:b  =  a':br  and  let  r 
be  ad  justed  until  there  is  no  current  in  the  galvanometer,  then 

x:  r  =  a:  b. 

For  let  the  currents  be  as  indicated  in  Fig.  56.  Since  D  and  G 
are  at  the  same  potential,  the  sum  of  the  steps  of  potential  around 
DAEG  must  equal  zero,  and  the  same  must  be  true  of  the  sum 
of  the  steps  of  potential  around  DCFG.  Hence, 

—  ai-\-xI-\-a'ir  =  o 


COMPARISON   OF    RESISTANCES.  173 

Divide  the  first  by  a  and  the  second  by  b  and  add,  remembering 
that  a': a  =  b':b.    Then 

X   T          r  T 

-/— T/*o. 

a       b 

Hence 

x  :a  =  r  :b. 

The  form  of  Double  Bridge  made  by  Hartmann  and  Braun  is 
very  satisfactory.  The  correspondence  of  parts  to  parts  of  the 
diagram  is  readily  traced.  The  ratio  of  a  to  b  can  be  varied  from 
100  to  100  to  10  to  10,000;  moreover,  a  and  b  may  be  inter- 
changed and  so  the  ratfo  reversed.  Similar  remarks  apply  to 
a'  and  b1 '.  Thus  values  of  x/r  varying  from  10/10000  to  10000/10 
may  be  measured.  The  variable  r  may  be  varied  from  0.044 
down  to  o,  but  can  hardly  be  read  with  an  accuracy  of  i%  below 
o.ooi.  Hence  values  of  x  between  o.ooi/iooo  or  o.oooooi  and 
0.044  X  1000  or  44  may  be  measured  by  the  bridge. 

Care  must  be  taken  not  to  injure  the  standardized  bar  by 
scraping  the  contact  maker  along  it.  The  contact  maker  must 
be  raised  for  each  movement.  Do  not  allow  the  sharp  jaws  of 
the  clamps  to  come  down  on  the  bar  too  suddenly,  for  they 
might  cut  into  the  bar  somewhat. 

Test  as  many  as  possible  of  the  following  materials : 
(i)  Brass.     (2)   Iron.     (3)  Copper.     (4)  Zinc.     (5)  Lead.     (6) 
Carbon.     (7)  Rail  Bond, 
and  calculate  the  Specific  Resistance  of  each. 

Questions. 

1.  Considering  this  as  a  modified  fall  of  potential  method,  why  should  a, 
b,  a',  b',  be  of  very  large  and  E  B  F  of  very  small  resistance? 

2.  Does  the  battery  current  need  to  be  steady?    Why? 

3.  Could  an  alternating  current  be  used  in  any  circumstances? 


LII.     COMPARISON    OF  RESISTANCES   BY   THE    CAREY- 
FOSTER  METHOD. 

References — Henderson's  Electricity  and  Magnetism,  pp.   53-57;  Stewart  and 
Gee's  Practical  Physics,  II,  pp.  158-170;    Watson  (Pr.),  §181. 

To  find  very  accurately  the  difference  between  two  very  nearly 
equal  resistances  R  and  5,  connect  th?m  arid  two  other  nearly 
equal  resistances,  P  and  Q,  as  indicated  in  the  diagram,  where  ab 


ELECTRICITY  AND   MAGNETISM. 


is  a  very  uniform  wire,  which  we  shall  suppose  to  have  a  resist- 
ance of  more  than  I  ohm.  Let  the  resistance  of  unit  length  of  the 
wire  ab  be  p.  Let  the  distance  ad  be  Xi,  when  a  balance  has  been 
obtained  in  the  usual  way.  Then  exchange  R  and  S,  and  again 


FIG.  57 

obtain  a  balance.  Denote  the  new  value  of  ad  by  x2.  Since  P 
and  Q  have  not  been  changed  and  the  total  resistance  R,  S,  and 
ab  was  not  changed,  it  is  clear  that  R  +  xip  =  S  +  x2p,  or 
R  —  S  =  (x2  —  Xi)p.  To  find  the  value  of  p,  replace  Rby  a  stand- 


Fro.  58 

ard  I -ohm  coil,  and  S  by  a  heavy  connector  of  negligible  re- 
sistance, and  proceed  as  above;  then  p(x2  —  Xi)  =  i. 

The  exchange  of  R  and  6"  is  made  by  means  of  a  special  key 
designed  so  that  the  resistance  of  the  connections  will  remain  the 
same  (see  Fig.  58). 


BATTERY    RESISTANCE    BY   MANCE's   METHOD. 


175 


To  compare  a  box  of  unknown  errors  with  a  standardized  box, 
the  difference  between  each  resistance  of  the  former  and  a  cor- 
responding resistance  of  the  latter  is  found  by  the  above  method. 

To  calibrate  two  boxes,  put  one  in  place  of  R  and  replace  S  by 
a  standard  i-ohm  coil  and  so  find  exactly  the  value  of  each  i-ohm 
unit  in  the  box.  Then  replace  the  standard  by  the  other  box, 
in  position  S,  and  compare  the  i-ohm  units  of  the  second  box 
with  those  of  the  first  box.  Then  compare  a  2-ohm  unit  in  one 
box  with  two  i -ohms  in  the  other,  and  so  on.  Special  care  must 
be  taken  to  avoid  confusion  in  making  the  calculations,  and  for 
this  purpose  the  box  resistances  may  be  denoted  by  Ii,  I2,  Hi, 
II2,  etc.,  for  one  box,  and  I'i,  II'2,  etc.,  for  the  other. 

Questions. 

1.  State  the  formula  for  Wheatstone's  Bridge  before  and  after  R  and  S  are 
interchanged  and  therefrom  deduce  the  above  formula. 

2.  What  advantages  and  what  disadvantages  has  the  Carey-Foster  Bridge 
compared  with  the  Wheatstone  Bridge? 

3.  Do  P  and  Q  need  to  be  exactly  equal,  and  why? 

4.  Which  connecting  wires  must  be  of  low  resistance? 


LIE.    BATTERY  RESISTANCE  BY  MANCE'S  METHOD. 

References — Hadley,  p.  322;  Maxwell's  Electricity  and  Magnetism,  §357;  Schuster 
and  Lee's  Practical  Physics,  pp.  304-306;  Watson*(Pr.},  §195. 

The  resistance  of  a  battery  may  be  determined  by  placing  it  in 
the  ''unknown  arm"  R  of  a 
Wheatstone's  Bridge  (p.  145). 
In  this  case  there  will  be  a  cur- 
rent through  the  galvanometer 
when  the  bridge  battery  is  not 
connected.  But  if  P,  Q  and  S  be 
adjusted  until  there  is  no  change 
in  the  deflection  when  the  key  of 
the  bridge  battery  is  pressed,  the 
points  to  which  the  galvanometer 
is  connected  wilLbe  at  the  same 
potential  so  far  as  the  effect  of 
the  bridge  battery  is  concerned. 
Since,  when  the  adjustments  are 


FIG.  59. 


right,  the  bridge  battery  sends  no  current  through  the  galva- 


176  ELECTRICITY  AND   MAGNETISM. 

nometer,  this  battery  may  be  removed  and  the  key  alone  will 
serve  to  test  the  adjustment  of  P,  Q,  and  S. 

If  the  deflection  of  the  galvanometer  is  too  great  to  be  readable, 
the  control  magnet  (in  the  case  of  a  Kelvin  galvanometer)  may 
be  used  to  bring  the  needle  back,  or  the  galvanometer  may  be 
shunted  or  a  resistance  put  in  series  with  it.  Most  cells  vary 
slightly  in  resistance  and  e.  m.  f.  when  on  closed  circuit;  hence, 
the  keys  should  not  be  pressed  longer  than  is  necessary. 

In  Lodge's  modification  of  Mance's  Method  a  condenser  is 
placed  in  series  with  the  galvanometer.  There  will  then  be  no 
continuous  current  through  the  galvanometer;  but,  if  the  ad- 
justments of  P,  Q,  and  S  are  not  right,  on  pressing  the  key  by 
which  the  adjustment  is  tested  the  galvanometer  will  be  momen- 
tarily deflected. 

Questions. 

1.  Why  is  there  a  slow  movement  of  the  galvanometer  needle  when  the 
keys  are  kept  pressed? 

2.  Should  the  condenser  be  of  large  or  small  capacity?    Would  a  Leyden 
jar  do? 


LIV.    TEMPERATURE  COEFFICIENT  OF  RESISTANCE. 
SPECIFIC    RESISTANCE. 


References  —  Elementary:  Duff,  §§444-447;  Ames,  pp.  731-732; 

Kimball,  §647;  Reed  &  Guthe,  §278;  Spinney,  §285;   Watson,  §482.—  More 
Advanced:   Henderson's  Electricity  and  Magnetism,  pp.  95-101, 

The  resistance  of  most  solids  increases  as  the  temperature 
rises;  carbon  is  one  of  the  exceptions,  for  its  resistance  decreases. 
For  moderate  ranges  of  temperature  the  resistance  is  approxi- 
mately a  linear  function  of  the  temperature  or,  if  RQ  be  the  re- 
sistance at  o°  and  R  that  at  t°, 

R  =  R0(i  +  at) 

The  constant  a  is  called  the  temperature  coefficient  of  the  mate- 
rial. It  may  be  defined  as  the  change  per  ohm,  referred  to  the 
resistance  at  o°,  per  degree  change  of  temperature. 

The  change  of  resistance  can  be  most  conveniently  studied  by 
the  box  form  of  Wheatstone's  Bridge  (p.  145). 


TEMPERATURE   COEFFICIENT   OF   RESISTANCE.  177 

(A)  For  finding  the  temperature  coefficient  of  a  wire  such  as 
copper,  a  length  sufficient  to  give  several  ohms  resistance  should 
be  used.  The  determination  of  the  temperature  coefficient  does 
not  require  that  the  dimensions  of  the  specimen  should  be  known, 
but  the  specific  resistance  of  the  specimen  may  be  determined  at 
the  same  time.  Hence  the  length  and  mean  diameter  of  the  wire 
should  be  carefully  measured.  The  wire  should  then  be  soldered 
to  heavier  lead  wires  and  immersed  in  a  bath  of  oil,  and  its 
resistance  determined  at  intervals  of  about  10°  as  the  tempera- 
ture is  raised.  The  thermometer  should  be  placed  inside  the  coil 
so  as  to  be  as  nearly  as  possible  at  the  temperature  of  the  latter. 
It  will  be  an  improvement  if  the  coil  and  thermometer  are  in  a 
tube  that  is  immersed  in  the  bath,  the  opening  of  the  tube  being 
closed  with  cotton-wool. 

To  keep  the  temperature  constant,  while  measuring  the  resist- 
ance, would  be  difficult.  The  following  method  will  be  found 
to  give  much  better  results:  Having  measured  the  resistance  at 
the  temperature  of  the  room,  adjust  the  known  resistance  of  the 
bridge  so  that  there  would  be  a  balance  if  the  resistance  of  the 
wire  were  decreased  4  or  5  per  cent.  The  galvanometer  will  be 
deflected.  Now  heat  the  wire  very  slowly  and  the  galvanometer 
reading  will  begin  to  drift  toward  zero.  When  it  just  reaches 
zero,  read  the  thermometer  and  continue  the  process  step  by 
step.  At  the  close  of  the  experiment,  the  coil  should  be  discon- 
nected, the  ends  of  the  lead  wires  joined,  and,  with  suitable  ratio 
resistances,  the  resistance  of  the  leads  determined.  If  appreciable, 
it  should  be  subtracted  from  the  previously  observed  resistance. 

The  various  resistances  and  temperatures  should  then  be 
plotted  in  a  curve  that  should  be  approximately  a  straight  line. 
If  exactly  a  straight  line  is  obtained,  the  temperature  coefficient 
should  be  calculated  from  two  reliable  and  widely  separated 
points  on  the  curve.  Let  R  and  Rf  be  the  resistances  at  /  and  /', 
respectively.  Substituting  these  values  in  the  above  equation 
we  shall  get  two  equations  from  which  R0  can  be  eliminated. 

If  the  plotted  readings  give  a  distinct  curve,  the  resistance  must 
be  expressed  as  a  quadratic  function  of  the  temperature. 

R  =  R»(i  +at  +  bP) 

12 


178  ELECTRICITY   AND   MAGNETISM. 

From  three  points  of  the  curve  three  equations  may  be  written 
down  and  from  these  a  and  b  may  be  calculated. 

(B)  For  finding  the  temperature  coefficient  of  carbon  an  incan- 
descent lamp  may  be  used.  As  it  would  be  difficult  to  determine 
accurately  the  temperature  of  the  filament  in  the  exhausted  bulb 
by  the  preceding  method,  water  may  be  used  for  the  bath  and 
two  careful  determinations  of  the  resistance  made,  the  first  being 
while  the  water  is  at  about  the  temperature  of  the  room,  and  the 
other  when  the  water  is  boiling.  In  each  case  the  final  determin- 
ation of  the  resistance  should  not  be  made  until  the  temperature 
of  the  filament  has  become  constant,  as  is  indicated  by  its  resist- 
ance becoming  quite  constant.  The  leads,  where  they  are  im- 
mersed in  the  water,  should  be  carefully  insulated  with  tape. 

Questions. 

1.  Consider  possible  thermoelectric  effects   due   to   the   presence    of   the 
solder.    Would  such  effects  occur  if  the  specimen  and  lead  wires  were  of  differ- 
ent metal? 

2.  Briefly  describe  a  platinum  resistance  thermometer  and  draw  a  dia- 
gram to  show  how  it  is  used. 


LV.    SPECIFIC  RESISTANCE  OF  AN  ELECTROLYTE. 
TEMPERATURE    COEFFICIENT. 

References— Elementary :  Duff,  §465;  Crew,  §§388-395;  Hadley,  pp.  329-331; 
Kimball,  §§608-616;  Reed  &  Guthe,  §287;  Spinney,  §§328-331;  Watson, 
§§539-541. — More  Advanced:  Ewell's  Physical  Chemistry,  pp.  54-57;  Hen- 
derson's Electricity  and  Magnetism,  pp.  80-84;  Kohlrausch's  Physical 
Measurements,  pp.  316-321;  Watson  (Pr.),  §200. 

The  object  of  this  experiment  is  to  determine  the  specific  re- 
sistance of  an  electrolyte — for  instance,  solutions  of  copper  sul- 
phate of  different  concentrations.  The  box  form  of  Wheat- 
stone's  Bridge  is  most  suitable  for  the  purpose  (p.  145). 

A  steady  current  from  a  battery  and  a  galvanometer  to  deter- 
mine when  there  is  a  balance,  as  ordinarily  used  with  Wheat- 
stone's  Bridge,  cannot  satisfactorily  be  used  in  measuring  the 
resistance  of  an  electrolyte,  for  a  steady  current  produces  in  a 
short  time  polarization  at  the  electrodes.  This  polarization 
leads  to  too  high  an  estimate  of  the  resistance  of  the  electrolyte, 
for  when  no  current  flows  through  the  galvanometer,  the  three 


SPECIFIC    RESISTANCE   OF   AN    ELECTROLYTE. 


179 


other  arms  of  the  bridge  are  balancing  the  potential  difference 
necessary  to  overcome  the  true  resistance  of  the  electrolyte  plus 
the  potential  difference  required  for  overcoming  the  polarization 
potential  difference  at  the  electrodes.  This  difficulty  is  obviated 
by  using  the  rapidly  alternating  current  from  the  secondary  of  an 
induction  coil  instead  of  a  steady  current  from  a  battery- 
The  time  that  the  current  continues  in  one  direction  is  so  short 
that  no  appreciable  accumulation  can  form  at  the  electrodes  to 
produce  an  opposing  difference  of  potential.  An  ordinary  galvan- 
ometer would  not  be  affected  by  an  alternating  current,  but  a 
telephone  which  is  a  very  delicate 
detector  of  an  alternating  current 
may  be  substituted. 

In  the  simplest  form  of  appa- 
ratus a  vertical  glass  tube  of  known 
cross  section,  which  may  be  found 
by  calipers  (p.  13),  holds  the  elec- 
trolyte. The  electrodes  are  con- 
nected to  wires  that  pass  through 
the  stoppers;  the  upper  electrode 
can  be  raised  or  lowered  as  desired. 
The  resistances  corresponding  to 
two  different  distances  of  separa- 
tion of  the  electrodes  should  be 
determined.  From  the  difference 

we  get  the  resistance  of  a  column  whose  length  is  the  difference 
in  the  two  lengths  and  thus  eliminate  uncertainty  as  to  remaining 
polarization  of  the  electrodes  and  the  exact  ends  of  each  column. 

An  improved  form  of  apparatus  is  illustrated  in  Fig.  61.  The 
electrodes,  A,  B,  are  clamped  in  collars  C,  D.  The  variation 
in  their  distance  apart  is  measured  with  vernier  beam  calipers 
applied  to  c  and  d.  The  cross  section  of  the  glass  tube  is  deter- 
mined by  measuring  from  a  burette  the  volume  of  water  con- 
tained between  the  marks  a  and  b,  the  lower  end  of  the  tube 
being  closed  by  a  rubber  stopper.  The  outer  glass  E,  (a  Wels- 
bach  chimney)  contains  a  water  bath  and  a  stirrer  and  a  ther- 
mometer. The  water  may  be  heated  by  the  ring  burner  F. 
Measurement  should  be  made  of  the  resistance  of  samples  of 


FIG.  60. 


i  So 


ELECTRICITY   AND    MAGNETISM. 


©  c 


0  D 


several  assigned  solutions  of  different  concentrations  at  room 
temperature  and  of  one  solution  also  at  a  high  temperature. 
In  measuring  resistances  it  will  probably  be  desirable  to  use 

equal  resistances,  e.  g.,  100  ohms,  in 
the  ratio  arms  of  the  bridge.  It  may 
be  impossible  to  obtain  a  balance  for 
which  there  is  no  sound,  for  even 
though  there  were  a  balance  for 
steady  current,  there  would  not  in 
general  be  a  balance  for  varying 
currents  such  as  are  used  in  this 
experiment,  owing  to  the  inductive 
electromotive  forces  of  capacity  and 
self-induction  in  the  resistance  coils. 
When  there  is  uncertainty  as  to 
whether  a  small  resistance  should  be 
added  or  cut  out,  the  ear  is  often 
assisted  by  adding  and  cutting  out 
a  larger  resistance  about  which  there 
is  no  doubt.  On  comparing  the 
change  of  tone  on  a  variation  of  this 
latter  resistance  with  the  variation 
of  tone  with  the  uncertain  resistance, 
one  can  often  decide  whether  the 
small  resistance  should  be  added  or 
not.  With  a  little  practice  one  should 
determine  resistances  within  I  per 
cent. 

Calculate  the  specific  resistance  of 
each  solution  at  each  temperature 
and  tabulate  the  results.  Find  also  the  temperature  coefficient 
of  the  solution  which  was  heated  and  calculate  its  specific  resist- 
ance at  o°. 

Questions. 

1.  Why  should  we  expect  the  resistance  to  decrease  with  increased  tem- 
perature? 

2.  What  is  supposed  to  be  the  nature  of  electric  conduction  in  an  electrolyte? 

3.  Are  the  specific  resistances  inversely  proportional  to  the  concentrations? 
Explain. 


3 


FIG.  61. 


COMPARISON  OF  E.  M.  F/S  BY  HIGH-RESISTANCE  METHOD.     l8l 

LVI.    COMPARISON  OF  E.  M.  F.'S  BY  HIGH-RESISTANCE 

METHOD. 

The  readiest  method  of  comparing  the  electromotive  forces 
of  cells  is  by  means  of  a  galvanometer  of  sufficiently  high  re- 
sistance. If  the  deflections  are  (by  the  use  of  added  resistances) 
kept  small  the  deflections  of  the  galvanometer  will  be  closely 
proportional  to  the  currents  that  pass  through  it  or  i  =  k.d 
where  k  is  a  constant.  Two  methods  may  be  employed  for  com- 
paring two  cells.  In  the  first,  called  the  "equal  resistance" 
method,  the  total  resistance  R  is  kept  constant  (the  resistance 
of  the  cells  being  supposed  negligible).  Hence,  by  Ohm's 
Law,  the  e.  m.  f.'s  are  proportional  to  the  currents,  that  is,  to 
the  deflections,  or 

Ei  =  di 

E2     d* 

In  the  other  or  "equal  deflection"  method,  such  resistances  are 
used  in  the  circuit  that  the  cells  cause  equal  deflections  of  the 
galvanometer.  Hence  by  Ohm's  Law,  since  the  currents  are 
equal,  the  electromotive  forces  must  be  proportional  to  the  re- 
sistances, or 


Both  methods  should  be  employed  to  find  the  e.  m.  f.'s  of  sev- 
eral cells  by  comparing  them  with  that  of  a  standard  Daniell 
cell  (p.  152).  Directions  for  the  adjustment  of  the  telescope 
and  scale  are  given  on  p.  23. 

(A)  Equal  Resistance  Method.  —  Make  R  such  that  the  stand- 
ard Daniell  cell  gives  a  deflection  of  about  5  cm.  on  a  scale  about 
50  cm.  from  the  mirror.  Make  a  reading  of  the  zero;  i.  e.,  when 
no  current  passes  through  the  galvanometer.  Send  the  current 
through  the  galvanometer  and  read  the  division  now  on  the  cross- 
hair. It  is  better  to  connect  the  cells  so  that  the  deflections  are 
always  on  the  same  side  of  the  zero.  Repeat  the  zero  reading  as  it 
is  liable  to  change.  In  reading,  use,  if  necessary,  the  method  of 
vibration  (see  p.  27).  If  the  vibrations  are  irregular  on  account 
of  trolley  currents  or  other  disturbances,  estimate  the  position 
of  equilibrium  from  the  vibrations  without  actually  making 


1 82  ELECTRICITY  AND   MAGNETISM. 

readings.  With  some  galvanometers  the  damping  is  so  great 
that  the  system  comes  to  rest  instead  of  vibrating  about  the 
position  of  equilibrium.  In  this  case  the  true  reading  can  be 
made  at  once.  Always,  if  possible,  estimate  tenths  of  the  small- 
est divisions.  When  you  have  thus  found  the  deflection  for 
the  standard,  find  similarly  the  deflection  for  as  many  different 
types  of  cells  as  time  allows.  The  internal  resistance  of  the  dif- 
ferent batteries  varies,  but  the  resistance  of  the  circuit  should  be 
so  high  that  the  differences  are  negligible  compared  with  the 
total  resistance  of  the  circuit.  The  galvanometer  may  be  shunted 
if  necessary.  Express  in  volts  your  final  values  of  the  e.  m.  f.'s 
of  the  cells  tested. 

(B)  Egital  Deflection  Method. — With  a  resistance  which  gives 
a  deflection  of  about  5cm.,  read  the  deflection  given  by  the 
standard  Daniell  cell.  Replace  the  standard  by  one  of  the  cells 
to  be  tested  and  vary  the  resistance  of  the  circuit  until  the  de- 
flection is  the  same  as  you  found  it  on  this  side  for  the  standard. 
Similarly  find  the  resistance  which  will  make  the  deflection  on 
the  other  side  the  same  as  that  given  by  the  standard  on  that  side. 
The  total  resistance  of  the  circuit  should  be  so  great  that  the  re- 
sistances of  the  cells  may  be  neglected;  but  it  will  probably  be 
necessary  to  take  the  resistance  of  the  galvanometer  into  ac- 
count. Take  the  mean  of  the  two  resistances  determined  above, 
plus  the  resistance  of  the  galvanometer,  as  the  resistance  re- 
quired to  give  the  same  deflection  as  the  standard  cell  gave  through 
the  box-resistance  used  with  it,  plus  the  galvanometer  resistance. 
The  resistance  of  the  galvanometer,  G,  must  be  determined  as  in 
Exp.  XLV  (last  paragraph). 

In  determining  the  possible  error  of  your  results,  estimate  the 
possible  error  of  resistances  from  the  least  change  in  resistance 
which  will  have  an  appreciable  effect,  and  the  possible  error  of  de- 
flections from  the  mean  deviation  from  the  mean  in  your  readings. 

Questions. 

1.  What  are  the  advantages  and  disadvantages  of  the  type  of  galvano- 
meter used  in  this  experiment  compared  with  other  types  used  in  the  labor- 
atory? 

2.  Which  of  the  two  methods  do  you  consider  the  better?    Why? 

3.  How  could  this  method  be  used  for  finding  the  internal  resistance  of 
a  cell? 

4.  Are  the  deflections  of  a  galvanometer  strictly  proportional  to  the  cur- 
rents?    Why? 


COMPARISON  OF  E.  M.  F.  S  BY  CONDENSER  METHOD. 


183 


LVII.    COMPARISON  OF  E.  M.  F.'S  AND  MEASUREMENT 

OF  BATTERY  RESISTANCE  BY  CONDENSER 

METHOD. 

References  —  Elementary:  Duff,  §§410-412,416,439,471-472;  Hadley,  pp.  168- 
170,  281-283;  Spinney,  §§261,  266;  Watson,  §§450-451.  —  More  Ad- 
vanced: Henderson's  Electricity  and  Magnetism,  pp.  185-187. 

When  a  condenser  of  capacity  C  is  connected  to  a  battery  of 
e.  m.  f.  E  it  receives  a  charge  Q=  CE.  If  it  be 
then  connected  to  a  ballistic  galvanometer,  the 
throw,  d,  will  be  proportional  to  Q,  or  Q  =  K.d, 
where  K  is  a  constant.  We  shall  apply  this  to 
(A)  compare  e.  m.  f.'s  and  (B)  measure  the 
resistance  of  cells.  We  shall  describe  these 
separately,  but  in  practice  they  may  be  com- 
bined. 

(A)  Suppose  the  condenser  is  first  charged 
by  a  battery  of  e.  m.  f.,  EI,  and  the  deflection       ! 
when  connected  to  the  ballistic  galvanometer  FIG.  62. 

is  d\j  and  suppose  that  when  this  same  con- 
denser has  been  charged  by  a  battery  of  e.  m.  f.,  E2,  the  deflection 
is  d2  ;  then 


Use  a  key  with  an  upper  and  lower  contact.  The  condenser 
should  be  connected  to  the  battery  when  the  key  is  down  and 
to  the  galvanometer  when  the  key  is  up.  Be  very  careful 
never  to  connect  the  battery  directly  to  the  galvanometer. 
When  a  discharge  is  sent  through  a  ballistic  galvanometer,  the 
needle  swings  over  to  one  side  and  then  swings  back.  Observe 
the  reading  of  the  scale  on  the  vertical  cross-hair  of  the  telescope 
when  the  needle  stops  and  turns  back.  Always  in  such  work 
estimate  tenths  of  the  smallest  division.  Before  each  throw 
bring  the  needle  as  nearly  as  possible  to  rest.  The  zero  is  likely 
to  change;  therefore,  before  each  throw,  record  the  zero,  and,  after 
each  throw,  record  both  the  turning-point  and  the  difference  be- 
tween this  turning-point  and  the  zero,  i.e.,  the  amount  of  the  throw. 


184 


ELECTRICITY   AND    MAGNETISM. 


Always  charge  the  condenser  for  approximately  the  same 
length  of  time,  for  instance,  five  seconds.  With  a  standard 
Daniell  cell  (p.  152),  record  six  throws  on  one  side.  Reverse  the 


A  US 

\ 

\\  

V 

EIN 

FIG.  63. 


FIG.  64. 


battery  connections  and  record  six  throws  on  the  other  side.  Let 
the  mean  of  these  be  d\.  Replace  the  Daniell  cell  by  one  of 
another  type  and  find  as  before  the  mean  throw  d2.  If  £2  is  the 
e.  m.  f.  of  the  latter  cell 

£2=1.105   2 
di 

Measure  in  this  way  the  e.  m.  f.  of  as  many  cells  of  different  types 
as  possible. 

(B)  This  method  of  finding  the  resistance  of  a  cell  depends  on 
the  fact  that  when  the  poles  of  a  cell  of  resistance  B  are  joined 
by  a  conductor  of  resistance  r,  the  difference  of  potential  of  the 
poles  is  not  equal  to  the  e.  m.  f.,  £  of  the  cell,  but  depends  on  the 
relative  magnitudes  of  r  and  B.  For,  by  Ohm's  law  applied  to 
the  whole  circuit,  E  =  (B  -J-  r)i  and,  if  e  is  the  difference  of 
potential  of  the  plates,  by  Ohm's  law  applied  to  r  only,  e  =  ri. 
Hence,  e/E  =  r/(B  +  r).  If  the  deflection  of  the  galvanometer 
is  dl1  when  the  condenser  C  is  charged  by  the  shunted  cell  and  d 
when  the  same  cell  is  not  shunted,  Ce  =  Kdl,  and  CE  =  Kd. 
Hence  e/E  —  d\/d  =  r/(B  +  r)  and  so,  solving  for  B, 


B  =  r 


d-d1 


dl 


The  connections  are  the  same  as  when  comparing  e.  m.  f.'s 
with  the  addition  of  a  circuit  containing  a  resistance  and  a  very 
low  resistance  key  (e.  g.,  a  mercury  key),  connecting  the  poles 
of  the  cells.  The  battery  should  be  short-circuited  just  before 


MEASUREMENT    OF    POTENTIAL   DIFFERENCE.  185 

the  charging  key  is  depressed,  and  the  short-circuiting  key 
should  be  released  immediately  after  the  other,  otherwise  the 
battery  will  run  down.  Choose  such  a  short-circuiting  resistance 
that  the  galvanometer  throw  is  reduced  to  about  half  the  value 
which  it  has  without  the  short-circuit.  Do  not  use  the  plug-box 
resistances  for  this  work,  on  account  of  the  danger  of  burning 
them  out,  but  use  open  wound  resistances  of  large  wire.  Find 
the  internal  resistance  of  cells  of  several  different  types. 

In  estimating  the  possible  error  of  your  results,  estimate  the 
possible  error  of  your  mean  readings  from  the  mean  deviation 
from  the  mean  in  the  individual  readings. 

Additional  exercises  (to  be  performed  if  time  permit.) 

(C)  Study  the  effect  of  length  of  time  of  charge  by  means  of  the 
throws  obtained  with  the  condenser  charged  for  difierent  lengths 
of  time  with  the  same  battery. 

(D)  Study  the  leakage  of  the  condenser  by  comparing  the  throws 
when  the  condenser  has  been  successively  charged  with  the  same 
e.  m.  f.,  and  has  remained  charged  for  different  intervals  of  time. 

(E)  Study  the  electric  absorption  of  the  condenser  by  charging 
for  several  minutes,  discharging  and  reading   the  throw  and  im- 
mediately insulating;   after   one   minute,   again   discharge   and 
insulate.     Continue  this  process  for  several  minutes,  the  con- 
denser being  insulated  during  the  minute  intervals. 

Questions. 

1.  What  are  the  peculiarities  and  requirements  of  a  good  ballistic  galvan- 
ometer? 

2.  What  is  the  construction  of  a  condenser  and  what  do  absorption  and 
leakage  mean? 

3.  How  could  you  find  the  resistance  of  the  galvanometer  used,  employ- 
ing a  condenser  and  a  known  resistance? 

4  For  what  cells  is  (B)  unsuitable?     Explain. 

LVIII.   MEASUREMENT    OF    POTENTIAL  BY    POTENTIO- 
METER METHOD.    CALIBRATION  OF  VOLTMETER. 

References — Elementary:  Duff,  §495;  Ames,  pp.  674-675;  Hadley,  pp.  320- 
324. — More  Advanced:  Ayrton  and  Mathers'  Practical  Electricity,  §§174-180; 
Henderson's  Electricity  and  Magnetism,  pp.  200-208;  Watson  (Pr.),  §205. 

A  voltmeter  may  be  calibrated  by  balancing  a  part  of  the  e.  m.  f . 
applied  to  the  terminals  of  the  voltmeter  against  the  e.  m.  f. 
of  one  or  more  standard  cells.  To  do  this  a  very  high  resistance 


186 


ELECTRICITY   AND   MAGNETISM. 


circuit,  consisting  of  a  high  resistance-box  in  series  with  an  ordin- 
ary resistance-box,  is  placed  in  parallel  with  the  voltmeter.  The 
fall  of  potential  in  part  of  the  low  resistance-box  is  measured  by 
a  side-circuit  consisting  of  the  standard  cell,  a  sensitive  galvan- 
ometer and  a  key,  the  standard  cell  being  so  turned  that  it  tends 
to  send  a  current  through  the  galvanometer  in  the  opposite  direc- 
tion to  the  fall  of  potential  in  the  box. 

Let  e  be  the  e.  m.  f.  of  the  cell,  E  the  potential  difference  at  the 
terminals  of  the  voltmeter,  r\  the  resistance  across  which  the 
galvanometer  circuit  is  connected,  and  r%  the  remaining  resistance 
in  the  high-resistance  circuit.  When  r\  is  adjusted  so  that  there 
is  no  deflection  when  the  key  is  pressed 


r\ 

A  special  fuse-wire  for  very  low  currents  should  be  placed  im- 
mediately adjacent  to  the  battery  to  prevent  the  possibility  of 
injury  to  the  resistance-boxes.  The  main  circuit  should  be  closed 
through  a  spring-key  only  a  sufficient  length  of  time  to  enable 
the  voltmeter  of  the  galvanometer  to  be  read.  A  high  resistance 
should  be  placed  in  series  with  the  standard  cell  to  prevent  any 

considerable  current  passing 
through  it.  In  first  perform- 
ing this  experiment  it  is  well 
to  use  a  simple  and  inexpen- 
sive form  of  standard  cell, 
and  the  Daniell  cell  (p.  152) 
will  be  suitable.  For  later 
and  more  accurate  work,  either 
the  Clark  or  the  Weston  cell 
should  be  used.  By  varying 
the  number  of  cells  in  the 


FIG.  65. 


main  circuit  or  using  different  resistances  in  the  main  circuit, 
different  voltages  at  the  terminals  of  the  voltmeter  may  be  ob- 
tained. If  a  sliding  rheostat  of  considerable  resistance  be  used 
for  this  purpose  part  of  the  adjustment  for  a  balance  can  be 
rapidly  made  by  means  of  it. 

The  above  method  will  not  apply  if  the  voltmeter  is  to  be 


MEASUREMENT  OF  CURRENT  BY  POTENTIOMETER  METHOD.  187 

tested  at  voltages  less  than  the  e.  m.  f.  of  the  standard  cell.  In 
this  case,  an  inversion  of  the  connections  may  be  used.  Instead 
of  balancing  a  variable  part  of  the  voltage  against  the  e.  m.  f.  of 
the  cell  a  variable  part  of  the  e.  m.  f.  of  the  cell  is  balanced 
against  the  voltage.  A  little  consideration  will  indicate  the  nec- 
essary change  of  connection. 

Instead  of  the  above  temporary  arrangement  of  circuits,  a  Poten- 
tiometer, which  consists  essentially  of  the  several  circuits  with 
the  necessary  resistances  and  keys  in  permanent  connection,  may 
be  used.  With  its  aid  the  work  may  be  performed  more  rapidly. 

A  simple  form  of  potentiometer  consists  of  a  long  wire  wound 
on  an  insulating  cylinder,  the  whole  wire  corresponding  to  the 
sum  of  r\  and  rz  (Fig.  65)  while  a  sliding  contact  gives  the  adjust- 
ment of  r\.  In  a  much  more  elaborate  form  of  potentiometer 
resistance  coilsf  of  such  (marked)  dimensions  are  used  that,  when 
the  regulating  rheostat  in  series  with  the  battery  has  been  adjusted 
to  give  a  balance  with  resistances  equal  numerically  to  (or 
ten  times)  the  e.  m.  f.  of  the  cell,  the  instrument  becomes  direct 
reading  for  any  other  cell.  By  means  of  a  step  down  'volt  box' 
(on  the  principle  of  Sect,  35,  p.  153)  decimal  fractions  of  high 
voltages  can  be  measured.  Its  parts  and  connections  should  be 
carefully  traced  out  with  the  assistance  of  a  large  diagram  (which 
may  be  attached  to  the  wall  near  the  instrument)  and  additional 
explanation  will  be  supplied  by  the  instructor. 

A  calibration  curve,  consisting  of  true  volts  plotted  against 
scale  readings,  should  be  drawn. 

Questions. 

1.  Prove  the  above  formula  by  applying  Kirchoff's  laws. 

2.  Draw  a  diagram  showing  the  connections  when  a  millivoltmeter  has  to 
be  calibrated. 

3.  Draw  a  diagram  to  show  how  the  above  method  could  be  adapted  to 
compare  the  e.  m.  f.'s  of  cells. 

LIX.   MEASUREMENT  OF  CURRENT  BY  POTENTIOMETER 
METHOD.    CALIBRATION  OF  AMMETER. 

References — Hadley,  p.  325;   Henderson's  Electricity  and  Magnetism,  p.  205; 

Watson  (Pr.},  §217. 

A  method  somewhat  similar  to  that  used  for  the  voltmeter 
may  be  employed.  The  current  from  a  storage  battery  that 


188 


ELECTRICITY   AND    MAGNETISM. 


passes  through  the  ammeter  passes  also  through  a  conductor 
of  large  current  capacity  and  of  measured  resistance  and  a  switch. 
The  potential  difference  at  the  ends  of  this  conductor  is  found 
by  a  shunt  circuit,  consisting  of  a  high  resistance-box  in  series 
with  a  box  containing  low  resistances.  In  parallel  with  the  latter 
is  a  circuit  containing  a  Daniell  cell,  a  sensitive  galvanometer 
and  a  key.  A  special  very  fine  fuse-wire  should  be  used  in  series 
with  the  two  boxes,  and  its  resistance  should  be  known  and  taken 
account  of  in  calculating  the  current.  The  large  conductor 
should  be  immersed  in  oil  and  its  temperature  kept  as  nearly  as 
possible  at  the  temperature  at  which  its  resistance  is  determined. 
To  prevent  heating,  the  main  current  should  be  closed  for  short 
intervals  only. 

The  galvanometer  should  be  protected  by  a  shunt  during  the 
first  adjustments.  Notice  first  in  which  direction  the  galvan- 
ometer moves  when  the  key  in  its  circuit  is  depressed.  The 
deflection  should  be  reversed  or  reduced  when  in  addition  the 

switch  is  closed.  If  this  is  found 
not  to  be  so,  the  connection  of 
either  the  Daniell  cell  or  the  stor- 
age batteries  should  be  reversed. 
The  resistance  in  the  box  nearest 
the  galvanometer  and,  if  neces- 
sary, in  the  other  box  also,  should 
be  varied  until  there  is  no  deflec- 
tion if  the  galvanometer  key  is 
depressed  when  the  switch  is 


When  the  adjustment  has  been 
obtained  as  closely  as  possible, 
the  fall  of  potential  between  the  points  of  the  high-resistance 
circuit  to  which  the  standard  cell  circuit  is  attached  equals  the 
e.  m.  f.  of  the  cell  (p.  152).  From  this  and  the  resistances  of  the 
boxes  the  fall  of  potential  between  the  ends  of  the  large  conductor 
is  found  and  then  from  the  resistance  of  the  large  conductor  the 
current  through  it  and  the  ammeter  is  calculated.  The  total 
resistance  in  the  two  boxes  must  be  kept  high.  A  preliminary 
calculation  will  show  about  how  large  the  resistance  of  large 


COMPARISON   OF   CAPACITIES   OF   CONDENSERS.  189 

capacity  should  be.  A  number  of  currents  distributed  over  the 
range  of  the  ammeter  should  be  used  and  from  the  results  a 
calibration  curve  should  be  drawn. 

In  the  above  we  have  assumed  that  the  voltage  applied  to  the 
conductor  exceeds  that  of  the  cell.  If  the  reverse  is  the  case, 
the  arrangement  must  be  inverted ;  i.e.,  part  of  the  e.  m.  f .  of  the 
cell  must  be  balanced  against  the  voltage  applied  to  the  con- 
ductor. This  method  must  be  applied  when  the  current  is  less 
than  the  quotient  of  the  e.  m.  f.  of  the  cell  and  the  resistance  of 
the  conductor. 

Instead  of  the  arrangements  of  circuits  above,  the  Potenti- 
ometer referred  to  in  Exp.  LVIII  may  be  used  to  measure  the 
fall  of  potential  in  the  conductor  of  large  current  capacity. 

Questions. 

1.  Why  must  the  resistance  in  the  shunt  circuit  be  large?     Calculate  the 
minimum  allowable  resistance  in  your  experiment. 

2.  Storage  batteries  giving  an  e.  m.  f.  of  50  volts  are  available  for  calibrat- 
ing an  ammeter  whose  range  is  from  5  to  25  amperes  and  whose  resistance  is 
0.2  ohms,     (a)  Using  the  connections  of  Fig.  66,  what  is  the  least  possible  value 
for  the  resistance  of  large  capacity?     (b)  What  is  the  greatest  value? 


LX.    COMPARISON  OF  CAPACITIES  OF  CONDENSERS. 

References  —  Gray's  Absolute  Measurements  in  Electricity  and  Magnetism,  Chap. 
VIII;  Hadley,  pp.  331-334;  Henderson's  Electricity  and  Magnetism, 
PP-  235-241;  Watson  (Pr.),  §226. 

Two  or  more  condensers  are  to  "be  compared  by  three  methods. 

(A)  First  Method.  —  Each  condenser  is  charged  in  turn  by  the 
same  battery  and  then  discharged  through  a  ballistic  galvan- 
ometer. Let  the  capacities  of  the  two  condensers  be  C\  and  C%. 
The  charges  which  they  receive  when  connected  to  a  battery  of 
e.  m.  f  .  E,  are  ft  =  CiE,  and  ft  =  C2E.  Let  the  throws  of  the 
galvanometer  when  the  condensers  are  discharged  through  it  be 
di  and  d%,  respectively.  Then 


(Exp.  LVII).     The  connections  are  the  same  as  in  Exp.  LVII, 
with  the  addition  of  one  or  more  keys  to  charge  alternately  two 


190 


ELECTRICITY   AND   MAGNETISM. 


or  more  condensers.  If  either  deflection  be  too  small,  additional 
cells  should  be  added.  Storage  cells  may  be  used  if  connections 
are  made  through  special  very  fine  fuse-wires  to  protect  the  re- 
sistances. 

If  either  deflection  be  too  large  the  galvanometer  should  be  shunted  by  a 
known  resistance,  S.  Let  G  be  the  resistance  of  the  galvanometer  determined 
as  in  Exp.  XLV  (last  paragraph),  d'  the  throw  obtained  with  the  galvan- 
ometer shunted,  d  the  throw  which  would  have  been  obtained  without  the 
shunt,  q'  the  quantity  of  electricity  passing  through  the  galvanometer,  q"  the 
quantity  passing  through  the  shunt.  Then 

£  € 

q'~S 

for  charges  of  electricity,  like  steady  direct  currents,  divide  inversely  as  the 
resistances  (p.  151).  Hence 


Since 


(B)  Bridge  Method. — The  two  condensers  to  be  compared, 
and  C2,  form  two  arms  of  a  Wheatstone's  Bridge,  two  high 

non-inductive  resistances,  RI 
and  Rz  (see  figure),  prefer- 
ably several  thousand  ohms, 
forming  the  other  two  arms. 
These  two  resistances  are 
adjusted  until  on  closing  the 
battery  circuit  at  a  the  gal- 
vanometer is  not  disturbed. 
Then  during  both  charge  and 

discharge  the   farther  poles 

Q    -r-  of  the  condenser  (A  and  B) 

. T     ,  must    remain    at    the   same 

FIG<  67.  potential  as  well  as  the  nearer 

poles  (joined  at  D).     Hence 

the  charges  Q\  and  Qz  in  the  condensers  must  have  the  ratio, 
Qi :  Qi : :  Ci :  €2-     But  the  quantities  which  have  flowed  into  the  con- 


COMPARISON    OF   CAPACITIES   OF   CONDENSORS. 


191 


densers  will  be  inversely  proportional  to  the  resistances  through 
which  the  charges  have  flowed,  that  is,  QiiQz'.'.RziRi-     Hence 


The  battery  key  has  an  upper  and  lower  contact;  the  upper 
contact  (b),  against  which  the  lever  ordinarily  rests,  short- 
circuits  the  battery  terminals  of  the  bridge,  thus  keeping  the 
condensers  uncharged.  The  sensitiveness  may  be  increased  by 
increasing  the  number  of  cells  in  the  battery,  and  also  by  using 
a  double  commutator  (see  p.  153).  Instead  of  a  galvanometer 
a  telephone  may  be  used  in  this  method,  the  battery  being  re- 
placed by  a  small  induction  coil. 

(C)  Thomson's  Method  of  Mixtures.  —  The  connections  are  as 
shown  in  the  figure.  KI  is  a  Pohl's  commutator,  K%  an  ordi- 


nary single  contact  switch.  When  the  swinging  arm  of  the  com- 
mutator is  in  the  position  aa',  the  two  condensers  are  charged, 
Ci  to  the  difference  of  potential  at  the  extremities  of  RI,  Cz  to 
the  difference  of  potential  at  the  extremities  of  R%.  The  swinging 
arm  of  the  commutator  is  now  placed  in  the  position  b'b'  and  the 
two  charges  are  allowed  to  mix.  If  they  are  exactly  equal, 
being  of  opposite  sign,  the  galvanometer  will  not  be  affected  when 
K2  is  depressed.  R!  and  R2  (which  are  large  resistances,  pref- 


192  ELECTRICITY   AND   MAGNETISM. 

erably  several  thousand  ohms),  are  adjusted  until  this  is  secured. 
The  charges  being  equal,  C\V\  =  C2V2,  and  since  Vi'.Vz'.'.Ri'.Rz 


C2     ~Ri 

Questions. 

1.  What  is  the  composite  capacity  of  three  microfarad  coadensers  in  par- 
allel?    In  series? 

2.  Which  of  these  three  methods  do  you  consider  the  best? 

3.  State  briefly  in  words  (without  formulae)  why  charges  divide  like  steady 
currents;  i.  e.,  inversely  as  the  ohmic  resistances. 

4.  Why  cannot  series  resistance  be  used  in  (A)  to  reduce  the  sensitiveness 
of  the  galvanometer. 

5.  Illustrate  the  principle  of  the  bridge  method  (using  sine  e.  m.  f.  and  tele- 
phone) by  a  vector  diagram  and,  if  possible,  derive  the  equation  for  the  re- 
lations in  the  diagram. 


LXI.   ABSOLUTE  MEASUREMENT  OF  CAPACITY. 

The  magnitude  of  a  capacity  can  also  be  found  without  the  use 
of  a  known  capacity  with  which  to  compare  it.  This  can  be 
done  in  different  ways.  The  following  is  one  of  the  simplest. 

Let  Q  be  the  charge  received  by  the  condenser  of  unknown 
capacity  C  when  connected  to  a  cell  of  known  e.  m.  f.,  E.  Then 


To  find  Q  discharge  the  condenser  through  a  ballistic  galvan- 
ometer the  constant  of  which  has  been  found  by  the  method  of 
Exp.  LXIV.  If  the  constant  be  K  and  the  deflection  D,  Q  =  KD. 

LXII.    COEFFICIENTS  OF  SELF-INDUCTION  AND  OF 
MUTUAL  INDUCTION. 

References — Elementary:  Duff,  §§509-510;  Ames,  pp.  743-745;  Hadley,  pp. 
417-422;  Kimball,  §727;  Reed  &  Guthe,  §333;  Spinney,  §§364,  365; 
Watson,  §518. — More  Advanced:  Watson  (Pr.)f  §§231-235. 

The  coefficient  of  self-induction  of  a  circuit  is  the  number  of 
magnetic  lines  of  force  which  link  with  the  current  when  the 
circuit  is  traversed  by  unit  current.  Owing  to  the  difficulty  of 
calculating  this  important  quantity  from  the  dimensions  of  the 
circuit,  experimental  methods  of  determination  have  much  value. 


COEFFICIENTS   OF   INDUCTION. 


193 


(A)  Probably  the  best  method  (using  direct  currents)  is 
Anderson's  Modification  of  Maxwell's  Method.  The  connections 
are  shown  in  the  figure.  The  coil  of  self-induction  L  and  re- 
sistance Q  is  made  one  arm  of  a  Wheatstone's 
Bridge  (preferably  Post-office  Box  form).  Ob- 
tain a  balance  for  steady  currents  by  proper 
variation  of  5,  so  that  when  K\  is  closed  and 
then  K-i,  the  galvanometer  is  not  disturbed. 
For  delicacy  of  adjustment  it  is  well  to  either 
have  a  resistance  which  can  be  varied  continu- 
ously form  a  part  of  S,  or  make  the  ratio  arms 
P  and  R  such  that  S  is  large.  Vary  r,  the  re- 
sistance-in  the  battery  circuit,  and  if  necessary, 
vary  the  capacity  of  the  condenser  C  until  there 
is  a  balance  for  transient  currents;  i.  e.,  until 
the  galvanometer  is  not  disturbed  when  K%  is 
depressed  and  K\  depressed  afterward.  Then  if  C  is  the  capacity 
of  the  condenser. 

L  =  C[r(Q+P)+P.S\. 

For  at  time  Met  x  =  current  in  branch  AB,  y  =  current  in  AD  =  cur- 
rent in  D  E,  z  =  current  in  BE  .'.  current  in  r  =  y  +  z.  q  =  charge  in  con- 
denser, e  =  potential  difference  of  its  poles  =  Rz  +  r  (y  -f  2). 

Since  there  is  a  balance  for  transient  currents  we  may  equate  the  e.  m.  f. 
in  AD  to  that  in  AB.  Hence 

^=p*. 


FIG.  69. 


The  current  in  the  branch  containing  the  condenser  is  (x— z);  but  it  can  also 
be  expressed  as 

—  or  C— 
Hence 


Now  since  there  is  a  balance  for  steady  currents  RQ  =  PS  and  since  Rz  = 
Sy,  it  readily  follows  that 


If  the  resistances  are  expressed  in  ohms  and  the  capacity  in 
farads,  the  results  will  be  in  henries. 

The  deflection  due  to  transient  currents  will  generally  be  very 
13 


194 


ELECTRICITY   AND    MAGNETISM. 


small  and  care  must  be  taken  (i)  they  are  not  masked  by  a 
slightly  imperfect  balance  for  steady  currents  and  (2)  that  the 
mere  absence  of  a  deflection  which  may  be  due  to  insufficient 
sensitiveness,  wrong  connections,  etc.,  is  not  mistaken  for  a 
balance.  The  only  true  test  of  a  balance  is  a  reversal  of  the 
direction  of  the  galvanometer  deflection  in  passing  from  too 
small  to  too  large  a  value  of  the  variable  resistance. 

Measure  the  self-induction  of  a  coil  whose  length  is  great  com- 
pared with  its  diameter  and  compare  the  result  with  that  calcu- 
lated. To  calculate  the  coefficient  of  self-induction  it  is  necessary 

to  know  the  number  of  lines  of 
force  passing  through  the  coil. 
This  number  multiplied  by  the 
number  of  turns  will  give  the 
number  which  link  with  the 
current;  i.  e.,  the  self-induc- 
tion. If  A  be  the  area  of  the 
cross-section  of  a  solenoid  of 
practically  infinite  length,  with 
_  UQ  turns  per  cm.  of  length,  the 

I number  of  lines  is  ^n^A  for  unit 

FlG  70  current.     The  number  of  turns 

in  a  length  d  is  nQd;  hence  the 

coefficient  of  self-induction  of  this  length  is  ^irAn^d'mC.  G.  S. 
units.  Reduce  to  henries  by  dividing  by  io9  (p.  154). 

To  secure  greater  sensitiveness  in  making  the  balance  for 
transient  currents,  replace  the  battery  by  the  secondary  of  a 
small  induction  coil  and  the  galvanometer  by  a  telephone,  or 
use  a  double  commutator  (see  p.  153). 

(B)  Comparison  of  Two  Coefficients  of  Self-induction. — The 
two  coils  of  self-inductions  LI,  Lz,  and  resistances  RI,  RZ,  are 
placed  in  two  arms  of  a  Wheatstone's  Bridge,  a  variable  re- 
sistance, r,  being  included  in  one  arm.  By  varying  r,  and,  if 
possible,  by  varying  one  of  the  self-inductances,  if  not,  by  vary- 
ing r,  P,  and  Q,  find  a  balance  for  both  steady  and  transient 
currents. 

Then  for  steady  currents 


R 


Q' 


COEFFICIENTS    OF    INDUCTION. 

and  for  transient  currents 


195 


Q' 


where  co  =  2ir  X  frequency.     Hence 


= 
L*     Q' 

The  above  adjustment  is  obtained  by  first  securing  a  balance 
for  steady  currents.     A  balance  for  transient  currents  is  then 
sought  by  varying  L\  or  L2.     If  this  cannot  be  secured,  r  and  P 
must  be  both  increased  or  decreased  and  a  balance  for  steady 
currents  again  obtained   and   then  one   for   transient  currents 
found  by  varying  LI  or  L2.     If  neither  LI  nor  L2  can  be  varied, 
a  balance  can  only  be  obtained  by  a  series  of  trials  as  above, 
the  ratio  of  P  and  (JRi  +  r)  being  kept 
constant,  so  that  the  steady  current 
balance  may  not  be  disturbed.    To  in- 
crease the  sensitiveness  with  transient 
currents,    a    double    commutator    (p. 
153)  may  be  used. 

(C)  The  coefficient  of  mutual  induc- 
tion of  two  coils  is  the  number  of  lines 
of  force  which  link  with  the  turns  of 
the  other  when  the  first  is  traversed 
by  unit  current.  Pirams  method  is 


FIG.  71. 


perhaps  the  most  satisfactory  for  the  experimental  determination 
of  coefficients  of  mutual  induction.  The  connections  are  shown 
in  Fig.  71.  If  M  be  the  required  coefficient,  C  the  capacity  of  the 
condenser,  and  r\  and  r2  the  values  of  the  variable  resistances 
for  which  the  galvanometer  is  not  disturbed, 

M  =  Cr\r^. 

For  let  the  steady  current  in  the  battery  circuit  =  i.  The  potential  dif- 
ference at  the  terminals  of  r\,  =  ir\.  The  charge  of  the  condenser  is  Cir\. 
If  /  =  time  required  to  establish  or  destroy  the  battery  current  the  average 
current  in  the  condenser  branch  during  this  time  =  Cir\lt  and  the  potential 
difference  at  the  terminals  of  r%  =  Cir \rzft.  Opposing  this  e.  m.  f.  in  the  gal- 
vanometer circuit  is  that  due  to  M,  the  average  value  of  which  =  Mi/t. 
If  the  galvanometer  is  not  disturbed  on  making  or  breaking  the  battery  cir- 
cuit, Cir\r-i./t  =  Mi/t.'.M  =  Cr\r^. 


196  ELECTRICITY   AND   MAGNETISM. 

The  secondary  of  M  is  acted  on  by  two  e.  m.  f.'s — one  due 
to  its  connection  with  the  main  circuit  in  which  there  is  an  e.  m. 
f.,  the  other  due  to  mutual  induction  between  the  primary  and 
secondary.  If  these  two  do  not  oppose  one  another  a  balance 
cannot  be  found.  If  such  is  found  to  be  the  case  the  connections 
of  the  secondary  to  the  galvanometer  circuit  must  be  reversed. 
To  increase  the  sensitiveness  of  the  method  a  double  commutator 
may  be  used  or,  better  still,  the  battery  and  galvanometer  may  be 
replaced  by  a  small  induction  coil  and  telephone. 

When  an  approximate  adjustment  has  been  found  C  and  r\ 
should  be  altered  until  the  sensitiveness  is  a  maximum,  and  then 
r\  and  r2  treated  in  the  same  way. 

Find  the  coefficient  of  mutual  induction  of  two  coils,  one  \vound 
upon  the  other,  and  one  of  which  is  long  compared  with  its 
diameter,  and  compare  the  result  with  that  calculated  from  the 
definition. 

Questions. 

1.  Coils  with  iron  cores  do  not  have  definite  induction  coefficients.     Ex- 
plain. 

2.  How  are  resistance  coils  in  boxes  wound  so  as  to  be  free  from  self-in- 
duction? 

3.  What  are  the  principal  difficulties  in  Method  (A)?     Can  you  suggest  a 
remedy? 


LXIII.    STRENGTH  OF  A  MAGNETIC  FIELD  BY  A 
BISMUTH  SPIRAL. 

References — Elementary:   Ames,  p.  758;    Hadley,  p.  296. 

The  electrical  resistance  of  a  bismuth  wire  is  changed  when  it 
is  placed  transverse  to  a  magnetic  field  and  the  magnitude  of  the 
change  depends  on  the  strength  of  the  field.  When  a  curve 
representing  the  resistance  of  a  flat  spiral  of  bismuth  as  a  function 
of  the  strength  of  the  magnetic  field  has  been  obtained  the  spiral 
may,  in  connection  with  a  Wheatstone's  Bridge,  be  used  to  meas- 
ure the  strength  of  any  magnetic  field  within  the  range  of  the 
calibration.  For  instance,  it  may  be  used  to  study  the  magnetic 
field  of  an  electromagnet.  The  following  three  points  may  be 
examined : 

(A)  Find  how  the  magnetic  field   between  the  poles  varies 


BALLISTIC    GALVANOMETER.  197 

when  the  strength  of  the  current  actuating  the  electromagnet  is 
varied  by  means  of  a  rheostat. 

(B)  Find  how  the  strength  of  the  field  midway  between  the 
pole-pieces  changes  when  the  distance  apart  of  the  pole-pieces 
is  varied,  the  current  being  kept  constant. 

(C)  Find  how  the  strength  of  the  field  in  an  equatorial  plane 
varies  with  the  distance  from  the  axis  of  the  pole-pieces. 

In  each  case  represent  the  results  by  means  of  a  curve. 


LXIV.    CONSTANT  AND  RESISTANCE  OF  A  BALLISTIC 
GALVANOMETER. 

References — Elementary:  Duff,  §439;  Ames.,  pp.  674,  712;  Hadley,  pp.  281-283. 
— More  Advanced-,  Ayrton  and  Mather's  Practical  Electricity,  §§145-149; 
Pierce,  Am.  Acad.  Arts  and  Sci.,  Vol.  42,  pp.  159-160;  Watson  (Pr.}, 
§§218-220. 

When  quantities  of  electricity  are  discharged  through  a  ballistic 
galvanometer  (p.  148)  the  throws  are  proportional  to  the  quanti- 
ties or, 

Q  =  K.d, 

where  K  is  the  constant  of  the  ballistic  galvanometer.  To  de- 
termine the  value  of  K  a  known  quantity  must  be  discharged 
through  a  galvanometer  and  the  throw  noted. 

This  known  quantity  might  be  obtained  from  a  condenser 
of  known  capacity,  charged  to  a  known  potential,  or  by  turning 
an  earth  inductor  (Exp.  XLIII)  in  a  field  of  known  strength. 
Both  of  these  methods,  however,  require  that  other  constants 
(capacity  and  e.  m.  f.  in  the  first,  strength  of  field  in  the  second) 
be  determined. 

A  simpler  method  is  to  use  a  so-called  calibrating-coil ;  i.  e., 
an  induction  coil  of  known  winding  without  a  magnetic  core. 
The  primary  is  a  long,  straight  helix,  so  long  that  there  is  no 
appreciable  leakage  near  the  center.  Over  the  center  there  is 
wound  a  secondary.  If  the  primary  be  of  n  turns  per  cm.  and 
the  secondary  be  of  n'  total  turns,  then  the  magnetizing  force 
produced  by  a  current  of  i  amperes  in  the  primary  is 


198  ELECTRICITY   AND   MAGNETISM. 

and  the  quantity  induced  in  the  secondary  by  making  or  breaking 
i  is 

hanf 


where  a  is  the  area  of  cross  section  of  the  cylinder  on  which  the 
primary  is  wound,  r  is  the  total  resistance  of  the  secondary 
circuit  and  the  factor  io8  is  required  when  q  is  in  coulombs  and 
r  in  ohms  (p.  154). 

The  number  of  lines  of  force  that  pass  through  the  secondary  is  a  h.    Hence 
when  h  is  increasing  the  induced  e.  m  f.  is  in  absolute  units. 

fdfak) 

The  quantity  induced  is  I  i  dt  and  i  equals  e/r.     Hence 

/•»'  d(ah)     n'ah 
4=      ~r      dr  =  T' 


From  the  above  expression  for  q  and  the  throw  d,  K  can  be 
calculated.  If  the  throw  is  small  it  may  be  doubled  by  reversing 
i  and  the  half  of  the  double  throw  taken  for  d.  Several  currents 
should  be  tried.  The  value  of  K  thus  found  is  in  coulombs  per 
scale  division. 

It  is  often  necessary  to  change  the  sensitiveness  of  a  ballistic 
galvanometer  by  shunting  it  or  putting  resistance  in  series  with 
it.  To  allow  for  this  we  must  know  the  resistance  of  the  gal- 
vanometer. 

The  resistance  of  a  ballistic  galvanometer  when  used  bal- 
listically  on  a  closed  circuit  is  different  from  its  resistance  when 
used  as  an  ordinary  galvanometer  for  steady  deflections.  This 
is  due  to  the  fact  that  the  galvanometer  coil  moves  in  a  magnetic 
field,  and  thus  an  induced  e.  m.  f.  is  produced.  Careful  tests  by 
Pierce  (see  references  above)  have  shown  that  this  has  the  same 
effect  as  an  added  resistance  and  that  this  apparent  addition 
to  the  resistance  of  the  galvanometer  remains  appreciably  con- 
stant over  a  wide  range  of  ballistic  deflections.  To  find  the 
effective  resistance  of  a  ballistic  galvanometer  we  may  use  the 
same  apparatus  and  connections  as  in  finding  the  constant  of  the 
galvanometer.  If  a  current  be  reversed  in  the  primary  of  the 


MAGNETIC    PERMEABILITY.  199 

calibrating  coil,  the  quantity  of  electricity  that  will  flow  through 
the  secondary  will  vary  inversely  as  the  total  secondary  re- 
sistance. Hence,  by  observing  the  throw  with  a  certain  primary 
current,  and  then  increasing  the  secondary  resistance  by  the 
insertion  of  a  box-resistance  and  repeating  the  reversal  of  the 
primary,  we  can,  by  proportion  (see  Exp.  XLV),  find  the  re- 
sistance of  the  galvanometer  when  used  ballistically. 

The  resistance  of  the  galvanometer  should  also  be  found  by 
the  method  of  Exp.  XLVI  or  that  of  Exp.  XLV  and  compared 
with  the  above. 


LXV.    MAGNETIC  PERMEABILITY. 

References — Elementary:  Duff,  §§489-496,  516;  Ames,  pp.  609,  615;  Hadley, 
pp.  384-392;  Kimball,  §§679-682;  Reed  &  Guthe,  §322;  Spinney,  §301; 
Watson,  §§502-504. — More  Advanced:  Ayrton  and  Mather's  Practical 
Electricity,  §§203-205;  Ewings  Magnetism  in  Iron,  Chap.  Ill;  Hender- 
son's Electricity  and  Magnetism,  pp.  282-284;  Watson  (Pr.},  §§239-240. 

A  current  in  a  long  solenoid  of  wire  will  produce  near  the 
center  of  the  solenoid  a  magnetic  force  H,  which  may  be  specified 
by  the  number  of  lines  of  force  per  unit  of  area  at  right  angles 
to  the  lines.  If  a  long  iron  rod  be  now  thrust  into  the  solenoid, 
the  number  of  lines  of  force  (now  called  lines  of  induction)  will 
be  much  greater,  say  B  per  unit  of  area.  The  permeability  of 
the  iron  is  defined  as  JJL  =  B  -f-  H. 

If  this  experiment  were  performed  with  comparatively  short 
iron  rods,  it  would  be  found  that  B  would  be  less  the  shorter 
the  rod.  One  consistent  way  of  explaining  this  is  to  consider 
the  free  poles  developed  at  the  ends  of  the  rod  when  magnetized. 
A  little  consideration  will  show  that  they  of  themselves  would 
produce  a  magnetic  force  in  the  space  occupied  by  the  iron,  this 
magnetic  force  being  opposed  to  the  original  megnetizing  force, 
and  so  we  may  say  that  the  effective  magnetic  force,  H,  is  the 
original  magnetic  force  diminished  by  the  demagnetizing  force 
of  the  poles.  It  is  this  effective  magnetic  force  that  we  should 
divide  into  the  induction  to  get  the  permeability.  The  calcula- 
tion of  the  demagnetizing  force  is  usually  difficult  and  uncertain, 
and  so  it  is  better  to  take  some  method  of  eliminating  it. 


200  ELECTRICITY   AND   MAGNETISM. 

One  such  way  is  that  implied  in  the  statement  at  the  outset, 
to  use  a  long  rod,  for  that  will  diminish  the  magnitude  of  the 
demagnetizing  force  at  the  center.  But  the  necessary  length 
makes  it  inconvenient  to  test  specimens  in  this  way.  Another 
method  is  to  join  the  ends  of  the  rod  by  a  heavy  yoke  of  iron, 
for  opposite  poles  developed  in  the  yoke  neuralize  the  effect  of 
the  poles  in  the  rod.  (This  is  one  way  of  stating  the  case. 
Another  way  is  to  say  that  the  yoke  carries  around  the  lines  of 
force.  A  third  way  is  to  say  that  the  yoke  diminishes  the  mag- 
netic resistance  of  the  circuit.)  The  difficulty  with  a  yoke  method 
is  in  getting  a  satisfactory  contact  between  yoke  and  rod.  A 
very  small  gap  will  result  in  the  neutralization  being  not  quite 
complete  (or  in  leakage  of  lines  of  force  or  in  magnetic  resistance)  . 

A  more  satisfactory  method  is  to  take  an  endless  specimen; 
i.  e.,  a  ring.  Then  there  are  no  free  poles  and  no  demagnetizing 
force.  On  the  ring  a  magnetizing  coil  of  N  turns  per  cm.  is 
wound.  When  a  current  of  /  amperes  passes  through  it,  the 
magnetizing  force  produced  is 

(i). 


10 


For  finding  the  value  of  B  a  secondary  coil  is  wound  on  the  ring 
and  put  in  series  with  a  ballistic  galvanometer.  Suppose  the 
iron  initially  free  from  magnetism.  The  setting  up  of  the  field 
B  produces  a  discharge,  Q,  of  electricity  through  the  secondary. 
If  A  be  the  area  of  cross  section  of  the  ring  and  N'  the  total 
number  of  turns, 


R  being  the  total  (ohmic)  resistance  of  the  secondary  circuit. 
The  factor  io8  is  not  necessary  if  Q  and  R  are  in  absolute  units. 
It  must  be  used  when  Q  is  in  coulombs  and  R  in  ohms  (p.  154). 
If  the  throw  of  the  galvanometer  is  D 

Q  =  K.D 

where  K  is  the  ballistic  constant  (Exp.  LXIV).  If  K  is  not 
known,  a  calibrating  coil  for  determining  it  should  be  included 


MAGNETIC    PERMEABILITY. 


201 


in  the  arrangement  of  the  apparatus.     From  the  above  formulae 
and  the  data,  H,  B  and  /-i  can  be  calculated. 

In  making  the  connections  for  the  practice  of  the  method,  it 
is  much  better  to  have  a  clear  understanding  of  the  plan  and 
purpose  of  each  part  and  to  proceed  systematically  than  to 
copy  the  connection  from  a  diagram.  In  the  first  place,  the 
secondaries  of  both  coil  and  ring  should  be  kept  permanently 
in  series  with  the  galvanometer.  Then  a  switch  is  to'  be  so 
arranged  that  the  current  can  be  passed  through  either  the  pri- 
mary of  the  calibrating  coil  or  that  of  the  ring.  A  suitable  rheostat 


FIG.  72. 

and  ammeter  are  needed  in  the  primary  circuit.  If,  as  the  pri- 
mary current  is  increased,  the  deflections  of  the  galvanometer 
become  too  great  to  be  read,  a  resistance  must  be  put  in  series 
or  in  parallel  with  the  galvanometer.  The  former  is  preferable. 
In  choosing  this  added  series  resistance,  it  is  well  to  so  choose  it 
that  the  whole  new  secondary  resistance  is  made  a  simple  multi- 
ple of  the  former  resistance.  If  this  is  done  the  throw  will  be 
reduced  in  the  proportion  in  which  the  resistance  is  increased, 
and  all  throws  may  be  reduced  to  what  they  would  have  been 
with  the  original  resistance  by  multiplying  the  actual  throw  by 
the  proportion  in  which  the  secondary  resistance  was  increased. 
For  methods  of  bringing  the  galvanometer  to  rest,  see  p.  149. 


2O2  ELECTRICITY   AND   MAGNETISM.  ~ 

Before  readings  are  begun  the  ring  should  be  demagnetized 
as  thoroughly  as  possible.  This  can  be  done  by  passing  an  alter- 
nating current  through  the  primary  and  reducing  it  from  a  large 
value  to  zero  by  means  of  a  rheostat,  or,  by  rapidly  commutating 
and  at  the  same  time  reducing  a  direct  current.  Also  at  each 
new  value  of  the  magnetizing  current,  before  readings  are  taken, 
the  commutator  should  be  reversed  several  times,  so  that  the  iron  may 
come  to  a  steady  cyclical  state.  Instead  of  attempting  to  get 
the  throw  on  making  the  primary  current,  the  double  throw  on 
reversing  the  current  is  taken  with  both  calibrating  coil  and  ring 
and  divided  by  2. 

At  least  three  throws  that  agree  well  should  be  read  for 
each  strength  of  the  primary  current.  The  magnetizing  current 
should  be  increased  at  first  by  small  steps  to  bring  out  the  char- 
acteristic features  of  the  curve  of  magnetization,  afterward  by 
larger  steps.  The  work  need  not  be  continued  after  the  read- 
ings begin  to  differ  in  a  much  smaller  proportion  than  the  suc- 
cessive magnetizing  currents,  for  this  shows  approaching  satura- 
tion. The  throw  at  break  of  current  should  also  be  carefully  noted 
as  a  means  of  estimating  the  permanent  magnetism;  for  from  the 
throw  at  break  the  diminution  of  B,  and,  therefore,  the  residual 
value  of  B,  can  be  calculated  as  above. 

In  the  report  the  various  values  of  /,  H,  Q,  B,  and  ju  should  be 
tabulated  and  a  curve  drawn  with  B  as  ordinates  and  H  as  ab_ 
scissae  (B-H  curve  or  curve  of  magnetization) .  On  the  same  sheet 
a  B-fj,  curve  should  also  be  drawn  and  a  third  curve  showing  the 
permanent  magnetism  as  deduced  from  the  throws  at  break  of 
the  current. 

Questions. 

1.  Why  is  only  the  ohmic  and  not  the  self-inductive  resistance  of  the  sec- 
ondary considered? 

2.  What  is  the  effect  of  the  windings  being  closer  together  on  the  inside 
of  the  ring  than  on  the  outside? 

3.  What  is  meant  by  intensity  of  magnetization?     Susceptibility?     Cal- 
culate a  few  values  from  your  results. 

4.  Define  "remanent"  magnetism  and  derive  a  formula  for  it  in  terms  of 
directly  measurable  quantities. 

5.  Could  the  throw  on  "make"  be  used  instead  of  the  double  throw  upon 
reversal,  the  iron  having  been  reduced  to  a  steady  state?    Explain. 


MAGNETIC   HYSTERESIS. 


203 


LXVI.    MAGNETIC  HYSTERESIS. 

References — Elementary:  Duff,  §497;  Hadley,  pp.  393-395;  Kimball,  §683; 
Reed  &  Guthe,  §323;  Watson,  §506. — More  Advanced:  Ayrton  and  Math- 
er's Practical  Electricity,  §§206-208;  Ewings'  Magnetism  in  Iron,  Chap.  V. 
Henderson's  Electricity  and  Magnetism,  p.  294. 

Let  a  magnetizing  force  applied  to  a  specimen  of  iron  as  in 
the  preceding  experiment  be  increased  step  by  step  and  let  the 
resulting  increases  of  magnetization  be  observed.  At  some  stage 
let  the  process  be  stopped  and  then  the  magnetizing  force  de- 
creased by  the  same  steps.  It  will  be  found  that  the  steps  of 
decrease  of  magnetization  are  less  than  those  by  which  it  at 
first  increased,  or  the  magnetization  lags  behind  the  magnetizing 
force.  This  is  called  hysteresis.  For  a  complete  view  of  the 


FIG.  73- 

process  a  cycle  must  be  completed,  i.  e.,  the  magnetizing  force 
must  be  decreased  step  by  step  to  zero,  then  increased  to  a  nega- 
tive value  equal  (numerically)  to  the  positive  value  at  which  the 
decreases  were  begun,  then  decreased  again  to  zero,  and  finally 
increased  again  to  the  highest  positive  value.  Thus  a  hysteresis 
loop  will  be  obtained. 

With  a  ring  specimen,  over  which  primary  and  secondary  coils 
are  wound,  there  are  two  methods  of  procedure. 

(A)  Step  by  Step  Method.— This  method  follows  closely  the 
general  description  given  above.  The  successive  steps  are  indi- 


204  ELECTRICITY   AND   MAGNETISM. 

cated  in  Fig.  73.  The  increases  or  decreases  of  I  must  be  made 
without  break  of  the  current.  The  steps  must  not  be  too  large 
or  the  points  on  the  curve  will  be  too  far  apart,  and  they  must 
not  be  too  small  or  the  work  will  become  tedious.  To  satisfy 
these  conditions,  place  in  the  primary  circuit  a  special  rheostat 
consisting  of  suitable  resistances  in  parallel,  each  of  which  can  be 
short-circuited  by  a  knife-edge  switch.  Such  a  rheostat  may  be 
made  up  with  resistances  permanently  connected  in  position, 
but  a  better  plan  is  to  use  removable  resistances.  In  the  latter 
case  a  considerable  collection  of  units  should  be  supplied,  and 
from  these,  by  a  preliminary  trial,  units  that  will  produce  suitable 
changes  of  /  (e.  g.,  from  4  to  0.5  amp.  by  steps  of  0.5  amp.)  should 
be  chosen  and  placed  in  position  in  the  rheostat. 

It  is  not  necessary  to  start  from  zero  magnetization.  Begin- 
ning with  the  highest  current  to  be  used,  reverse  several  times 
to  produce  a  cyclical  state  and  then  find  the  throw  on  reversal. 
From  this  the  maximum  value  of  B  can  be  calculated  as  in  Exp. 
LXV.  Then  diminish  the  current  by  steps  and  note  the  throw 
in  each  step.  After  the  step  that  reduces  the  current  to  zero, 
the  current  must  be  reversed  and  the  resistances  decreased  step 
by  step.  The  rest  of  the  process  needs  not  be  described.  From 
each  throw  the  corresponding  change  of  induction,  A  B,  is  calcu- 
lated as  in  Exp.  LXV.  When  the  cycle  has  been  completed  the 
algebraic  sum  of  the  throws  should  be  zero.  It  should  not  be 
necessary  to  change  the  sensitiveness  of  the  galvanometer;  it 
will  give  the  smaller  throws  with  less  accuracy,  but  they  are  less 
important.  This  "step  by  step "  method  of  measuring  hysteresis 
is  the  most  instructive  and  is  not  difficult  after  some  initial  prac- 
tice. It  has,  however,  the  disadvantage  that  an  error  in  one 
reading  of  the  galvanometer  vitiates  the  whole. 

(B)  The  Ewing-Classen  Method. — The  last-mentioned  dis- 
advantage is  avoided  in  this  method  by  starting  each  step  from 
the  maximum  value  of  B.  As  before,  we  first  find  by  reversals 
the  value  of  B  corresponding  to  the  maximum  value  of  /.  We 
then  diminish  I  (without  breaking  the  current)  and  from  the 
throw  we  calculate  the  diminution  of  B.  This  gives  us  a  second 
point  on  the  curve.  We  then  return  to  the  maximum  current 
and,  after  several  reversals,  to  re-establish  the  cyclical  state,  we 


MAGNETIC   HYSTERESIS. 


205 


again  decrease  7,  but  by  a  larger  amount  than  before.  From 
the  throw  we  again  calculate  the  .diminution  of  B  and  thus  get 
another  point  on  the  curve.  Proceeding  in  this  way,  we  reach 
the  stage  at  which  7  is  decreased  from  its  maximum  to  zero. 
This  gives  us  the  point  at  which  the  curve  crosses  the  axis  on 
which  B  is  plotted. 

A  simple  method  of  producing  the  above  changes  of  /  is  to 
connect  the  rheostat  described  under  (A)  in  parallel  with  one  of 
the  cross-bars  of  the  Pohl's  commutator  used  for  reversing  I 
(Fig.  74).  If  this  cross-bar  be  suddenly  removed,  the  resistance 


r 

FIG.  74. 


r 

FIG.  75- 


in  the  rheostat  will  be  thrown  into  the  circuit  without  breaking 
the  current. 

By  the  above  process,  we  have  obtained  that  part  of  the  de- 
scending branch  of  the  hysteresis  loop,  which  lies  to  the  right  of 
the  B  axis.  To  obtain  the  remainder  of  the  branch,  we  again 
proceed  by  steps  from  the  positive  maximum  value  of  B,  but, 
since  each  change  of  7  will  carry  it  from  its  positive  maxi- 
mum to  a  smaller  negative  value,  we  must  simultaneously 
diminish  and  reverse  the  current.  To  be  able  to  do  this,  remove 
the  cross-bar  of  the  commutator  which  is  in  parallel  with  the 
rheostat  and  turn  the  commutator  so  that  the  current  flows  to 
the  ring,  but  does  not  pass  through  the  rheostat  (Fig.  75).  If 
the  commutator  be  now  reversed,  the  current  will  be  reversed 
and  will  be  diminished  by  passing  through  the  rheostat.  Thus 
we  get  another  point  on  the  curve  and,  by  a  series  of  such  steps 


2O6  ELECTRICITY   AND    MAGNETISM. 

with  decreasing  resistances  in  the  rheostat,  the  descending  branch 
of  the  loop  is  completed.  To  trace  the  other  branch  we  might 
proceed  as  above,  beginning  each  step  from  the  negative  maximum 
of  B.  This,  however,  is  unnecessary,  since  we  would  evidently 
be  merely  repeating  the  previous  readings.  The  loop  is  sym- 
metrical about  the  origin,  and  the  co-ordinates  of  the  ascending 
branch  are  equal  to  those  of  the  descending  branch  but  with  signs 
reversed. 

It  can  be  shown  that  the  energy  expended  in  such  a  cyclical 
change  of  magnetization  is 

I 


ergs  per  c.c.  of  the  iron.  The  integral  also  represents  the  area 
of  the  loop,  due  allowance  being  made  for  the  scale  on  which  it 
is  plotted.  Hence  if  the  area  be  found  by  means  of  a  planimeter 
(the  use  of  which  will  be  explained  by  an  instructor),  the  energy 
loss  per  c.c.  per  cycle  can  be  calculated. 

The  total  number  of  lines  of  induction  through  each  turn  of  the  magnet- 
izing coil  is  AB.  Since  the  total  number  of  turns  is  IN,  when  B  is  being  in- 
creased there  is  induced  in  the  magnetizing  coil  an  e.  m.  f. 


£=_=-™       C.G.S.units. 

V  being  the  volume  of  the  core  (=  I  A}.     The  work  done  by  the  battery  in 
time  dt  in  overcoming  this  opposing  e.  m.  f.  is  4 

d  W  =  lEdt  =  IN  VdB  ergs 
TSTow  the  area  of  the  hysteresis  loop  is  the  integral  of  HdB  and 


V  r 

•-.w=-( 

A.TT  J 


HdB 


Questions. 

1.  What  rise  of  temperature  would  1000  cycles  produce  in  the  iron  if  no 
heat  were  lost? 

2.  How  much  less  would  the  energy  loss  be  if  the  maximum  magnetiza- 
tion were  half  as  great  as  in  your  cycle?     (According  to  Steinmetz's  law  the 
hysteresis  loss  is  proportional  to  51-8). 


THE  MECHANICAL  EQUIVALENT  OF  HEAT.         2O7 


LXVII.     (A)   THE  MECHANICAL  EQUIVALENT   OF  HEAT. 

(B)  THE  HORIZONTAL  INTENSITY  OF  THE 

EARTH'S  MAGNETISM. 

References — Elementary:  Duff,  §§436,  438,  466,  488;  Ames,  pp.  664-5,  688; 
Crew,  §§289,  350;  Hadley,  pp.  274,  336-340.  458;]  Kimball,  §§618,  654, 
692;  Reed  &  Guthe,  §§259,  269,  285;  Spinney,  §§318,  332,  349;  Watson, 
§§478,  493,  539.— More  Advanced:  Watson,  (Pr.),  §§211-212,214-215. 

If  Q  calories  of  heat  be  produced  in  a  conductor  by  the  passage 
of  a  current  i  for  time  /,  and  if  no  other  work,  chemical  or  me- 
chanical, be  performed,  then 

JQ  =  i*Rt, 

J  being  the  mechanical  equivalent.  If  i  be  expressed  in  amperes, 
R  in  ohms  and  Q  in  calories,  i2Rt  will  be  in  joules  (one  joule  being 
io7  ergs),  and  /  will  be  obtained  as  the  number  of  joules  in  a 
calorie. 

Q  can  be  measured  by  immersing  the  conductor  of  resistance 
R  in  a  known  mass  of  water  contained  in  a  vessel  of  known  water 
equivalent.  The  mass  of  water  may  be  obtained  with  sufficient 
accuracy  by  measuring-  it  from  a  burette.  To  reduce  the  effects 
of  radiation,  conduction  and  convection,  the  water  should  be  at 
the  beginning  of  the  passage  of  the  current  as  much  below  the 
temperature  of  the  room  as  it  finally  rises  above  it,  for  the  cur- 
rent is  kept  steady  and  the  temperature  of  the  water  therefore 
rises  steadily,  so  that  it  is  as  long  above  the  room  temperature 
as  below. 

But,  since  it  is  very  important  to  get  the  rise  of  temperature 
accurately  and  the  temperature  of  the  surroundings  is  somewhat 
indefinite,  the  initial  rate  of  warming  and  the  final  rate  of  cooling 
should  be  determined,  and  the  temperature  correction  calcu- 
lated as  explained  on  pp.  59-61. 

The  resistance  R  may  be  measured  against  a  standard  ohm  coil 
by  Wheatstone's  Bridge,  and,  since  it  will  be  found  necessary 
to  use  a  wire  of  comparatively  small  resista/ice,  R  should  be 
measured  with  great  care.  Leads  of  large  size  and  small  length 
should  be  employed  for  connecting  the  wire  to  the  bridge.  While 


208 


ELECTRICITY   AND    MAGNETISM. 


being  measured  it  should  be  immersed  in  the  calorimeter  in  water 
at  the  temperature  of  the  room,  so  that  the  mean  resistance 
throughout  the  experiment  is  obtained.  To  reduce  to  absolute 
units  the  resistance  in  ohms  is  multiplied  by  io9. 

The  current,  i,  may  be  obtained  from  its  chemical  effect  in 
another  part  of  the  circuit.  Careful  measurements  have  shown 
that  unit  current  (C.  G.  S.)  flowing  through  a  solution  of  copper 
sulphate  of  a  certain  strength  between  copper  electrodes  deposits 
0.00326  gms.  of  copper  per  second  on  the  cathode. 

The  form  of  copper  voltameter  employed  consists  of  a  glass 


Ca lor  i  nnetei 


Rheostat 

WWWVW 


Voltameter 
FIG.  76. 

vessel  containing  a  solution  of  copper  sulphate  into  which  dip 
three  plates.  The  two  outer  are  of  heavy  copper  and  are  both 
joined,  directly  or  indirectly,  to  the  positive  pole  of  the  battery, 
forming  the  anode.  The  intermediate  plate  is  thin  and  light, 
and  is  connected  to  the  negative  pole  of  the  battery,  forming 
the  cathode.  A  satisfactory  solution  consists  of  15  grams  of  cop- 
per sulphate  dissolved  in  100  grams  of  water,  to  which  are  added 
5  grams  of  sulphuric  acid  and  5  grams  of  alcohol.  (The  alcohol 
is  easily  oxidized, «thus  preventing  the  oxidization  of  the  deposit 
on  the  cathode  and  the  formation  of  polarizing  compounds  at  the 
anode.) 


THE  MECHANICAL  EQUIVALENT  OF  HEAT.        2OQ 

Clean  the  two  anode  plates  with  sand-paper  and  fasten  them 
in  the  two  outside  binding  posts  of  the  top  of  the  voltameter. 
Clean  with  sand-paper  a  cathode  plate,  wash  with  tap  water  and 
then  with  alcohol.  When  dry,  weigh  on  one  of  the  chemical 
balances,  weighing  to  milligrams  with  the  rider.  Wrap  in  paper 
and  set  aside.  Be  very  careful  not  to  touch  with  the  ringers 
any  part  of  the  plate  wrhich  will  be  in  the  solution,  after  it  has 
been  cleaned.  Clean  with  sand-paper  a  trial  cathode  and  mount 
it  on  the  middle  binding  post.  Before  putting  the  voltameter 
in  the  circuit,  dip  the  two  wires  which  are  to  be  connected  to  the 
voltameter  in  the  solution  of  the  voltameter.  Decrease  the 
variable  resistance  until  you  have  a  moderate  current,  but  do  not 
entirely  cut  it  out.  Notice  on  which  wire  copper  is  deposited 
as  a  brown  powder.  Connect  this  wire  to  the  cathode  of  the 
voltameter  and  the  other  wire  to  the  anode  plates. 

It  is  important  that  the  current  be  kept  constant.  It  is  true 
that  even  if  the  current  vary,  the  deposit  will  give  the  true  mean 
value  of  the  current.  But  what  is  needed  is  the  mean  value  of  i2, 
and  this  is  not  necessarily  the  same  as  the  square  of  the  mean 
value  of  i.  If  a  storage  battery  in  good  condition  be  used  as  the 
source  of  current,  the  current  will  not  vary  much;  nevertheless, 
a  tangent  galvanometer  or  an  ammeter  should  be  included  in  the 
circuit  to  test  the  constancy  of  the  current.  There  is  also  another 
reason  for  including  a  current  meter  of  some  form.  The  differ- 
ence of  potential  at  the  terminals  of  the  heating  coil,  or  iR,  must 
not  be  as  great  as  the  e.  m.  f.  (1.6  V)  that  will  electrolyze  water, 
otherwise  some  part  of  the  energy  of  the  current  will  be  spent 
in  chemical  work.  Knowing  R,  one  can  choose  a  safe  value  for  i. 
If  the  constant  of  the  galvanometer  be  not  known,  it  can  be  calcu- 
lated roughly  from  the  dimensions  of  the  coils  and  the  approxi- 
mate value  (say  0.18)  for  the  horizontal  component  of  the  earth's 
field  (see  Exp.  XLII),  and  so  the  deflection  corresponding  to  a 
safe  value  of  i  deduced.  An  ammeter,  if  available,  affords  a 
still  simpler  means. 

If  a  tangent  galvanometer  be  used  a  fairly  reliable  value  for  the 

horizontal  component  of  the  earth's  field  may  be  deduced  from 

the  results  of  the  experiment.     For  this  purpose  the  dimensions 

of  the  galvanometer  should  be  carefully  measured  and  the  cur- 

14 


210  ELECTRICITY   AND    MAGNETISM. 

rent  through  it  frequently  reversed  and  carefully  read.  The 
Helmholtz  form  of  tangent  galvanometer  may  be  used.  This  con- 
sists of  two  coils  separated  a  distance  equal  to  their  common 
radius,  with  the  needle  on  their  common  axis  midway  between 
them.  This  arrangement  of  two  coils  produces  a  very  uniform 
field  over  quite  an  area  where  the  needle  is  located,  allowing  the 
use  of  a  longer  needle.  The  formula  for  the  galvanometer  (the 
proof  for  which  will  be  found  in  text-books  on  physics)  is 


(If  a  simpler  type  of  tangent  galvanometer,  with  but  one  coil, 
is  used,  x  is  the  distance  from  the  plane  of  this  .coil  to  the  suspen- 
sion of  the  needle.) 

Equating  this  expression,  which  involves  the  dimensions  of 
the  galvanometer  and  the  deflection,  to  the  current  as  determined 
by  the  voltameter,  the  horizontal  component  is  deduced. 

If  an  ammeter  is  used  the  value  of  the  current  deduced  from 
the  copper  deposit  enables  us  to  test  the  accuracy  of  the  ammeter. 
This  is  in  fact  one  method  used  for.  the  absolute  calibration  of  an 
ammeter. 

When  the  adjustments  have  been  completed,  open  the  switch, 
remove  the  trial  cathode  and  put  in  place  the  other  cathode, 
which  has  been  kept  wrapped  in  paper.  Take  care  that  there 
is  no  metallic  connection  between  the  cathode  and  the  anode 
plates.  Remove  all  iron  from  the  neighborhood  of  the  tangent 
galvanometer  and  from  your  pockets.  All  wires  must  be  close  to- 
gether to  avoid  stray  induction  and  the  galvanometer  had  best 
be  at  some  distance  from  the  other  apparatus. 

After  reading  the  temperature  of  the  calorimeter  every  minute 
for  five  minutes  note  the  exact  second  on  an  ordinary  watch, 
and  close  the  switch.  As  soon  as  possible,  read  both  ends  of  the 
needle.  Reverse  the  current,  making  the  reversal  quickly,  and 
again  read  both  ends  of  the  needle.  Always  estimate  tenths. 
Reading  both  ends  of  the  needle  eliminates  error  due  to  the  axis 
about  which  the  pointer  turns,  not  coinciding  with  the  center  of 
the  graduated  circle,  and  reversing  eliminates  uncertainty  about 
the  reading  for  the  zero  position.  Keep  the  current  constant 


THERMOELECTRIC  CURRENTS.  211 

with  the  variable  resistance.  At  intervals  of  three  minutes 
(approximately)  read  both  ends  of  the  needle  and,  reversing, 
again  read  both  ends.  Read  the  temperature  every  minute  to 
tenths  of  the  smallest  division.  Allow  the  current  to  flow  until 
the  temperature  has  risen  to  the  extent  desired.  Note  the  exact 
second  of  breaking  the  circuit.  Continue  to  observe  the  tempera- 
ture at  minute. intervals  for  five  minutes.  Remove  the  cathode, 
being  very  careful  not  to  touch  the  copper  deposit.  Wash  it 
gently  with  tap  water  and  then  with  alcohol,  allowing  the  liquid 
to  simply  flow  over  the  surface.  When  it  is  dry,  weigh  as  before. 
Measure  very  carefully  the  diameter  of  the  coils  in  a  number  of 
directions,  and  from  the  mean  determine  r.  Count  n,  the  total 
number  of  turns  in  both  coils,  and  measure  2  x,  the  distance  be- 
tween the  centers  of  the  two  coils.  Weigh  the  inner  calorimeter 
vessel  and  note  of  what  metal  it  consists.  Plot  the  temperature 
readings  and  correct  for  radiation  (p.  59). 

Questions. 

1.  Calculate  the  exact  voltage  at  the  terminals  of  the  heating  coil. 

2.  What  sources  of  error  remain  uneliminated? 

3.  Calculate  the  mean  activity  of  the  current  in  the  coil  during  the  ex- 
periment. 

4.  What  are  the  peculiarities  and  advantages  of  the  tangent  galvanometer? 

5.  What  chemical  actions  take  place  in  the  voltameter?     To  what  is  the 
deposition  of  any  metal  proportional? 

6.  Why  is  it  advantageous  to  have  the  deflection  about  45°? 

7.  Why  must  the  current  be  constant? 


LXVIII.   THERMOELECTRIC  CURRENTS. 

References — Elementary:  Duff,  §§477-482;  Ames,  pp.  679-683;  Crew,  §369; 
Kimball,  §§664-670;  Reed  &  Guthe,  §§300-306;  Spinney,  §167;  Watson, 
§§498-499. — More  Advanced:  Hadley,  Chap.  XX;  /.  /.  Thomson's  Elements 
of  Electricity  and  Magnetism,  Chap.  XIV. 

To  the  ends  of  wires  of  copper,  nickel,  silver,  etc.,  lead  wires 
are  soldered  and  brought  to  binding  posts  on  a  board.  Below 
the  ends  of  the  board  are  vessels  containing  sand  or  oil  in  which 
two  test-tubes  are  supported.  The  junctions  are  placed  in  these 
test-tubes  as  indicated  in  Fig.  77.  The  binding  posts  are  con- 
nected, by  copper  wires,  to  a  key  of  as  many  parts  as  there  are 
wires  to  be  tested,  so  that  each  circuit  may  be  completed  through 


212  ELECTRICITY   AND   MAGNETISM. 

a  sensitive  galvanometer.  Thermometers  are  placed  in  the 
test-tubes  to  note  the  temperatures  as  one  vessel  is  being  heated 
by  a  burner. 

It  is  especially  important  that  the  temperature  should  be 
ascertained  accurately.  Hence  heat  should  be  applied  cautiously, 
especially  at  first,  and,  when  observations  are  to  be  made,  the 
source  of  heat  should  be  removed,  and  time  should  be  allowed  for 
the  temperatures  to  become  fairly  constant. 

The  galvanometer  reading  should  be  noted  with  the  greatest 
care  and  the  zero  should  be  frequently  tested.  After  each  read- 
ing of  the  galvanometer,  the  temperature  should  be  noted. 


Cu 

Cu 


Cu 


FIG.  77. 

If  a  high  resistance  galvanometer  of  sufficient  sensitiveness  is 
available,  the  other  resistances  may  be  neglected  and  the  various 

e.  m.  f.'s  will  then  be  proportional  to  the  deflections.     Or  a 
sensitive  low  resistance  galvanometer,  with  a  constant  high  re- 
sistance permanently  in  series,  may  be  used  with  similar  sim- 
plicity.    The   constant  of  the   galvanometer,   considered   as   a 
voltmeter,  may  be  found  by  applying  to  it  a  fraction  of  the  e.  m. 

f.  of  a  standard  cell  (pp.  86,  152,  153). 

With  this  arrangement  (which  will  be  readily  understood  from 
the  figure)  the  thermoelectric  force  of  each  circuit,  consisting  of 
lead  and  another  wire  (copper,  silver,  nickel,  aluminium,  tin), 
may  be  determined.  Curves  representing  the  results  should  be 
plotted  with  the  differences  of  temperature  of  the  junctions  as 
abscissae  and  the  e.  m.  f.'s  as  ordinates. 

If  a  low  resistance  galvanometer  of  low  sensitiveness  is  used, 
it  will  be  necessary  to  consider  it  as  an  ammeter.  In  this  case, 
the  resistances  of  the  various  circuits  and  of  the  galvanometer 
must  be  found  and  the  constant  of  the  galvanometer,  in  amperes 
per  unit  deflection,  must  be  obtained  by  connecting  it  in  series 
with  a  standard  cell  and  a  sufficient  known  resistance.  Thus, 


ELEMENTARY   STUDY   OF   ALTERNATING   CURRENTS.          213 

the   currents   and    the   resistances   being   known,    the   thermo- 
electric forces  can  be  calculated. 

For  the  study  of  a  case  of  thermoelectric  currents  over  a  wide 
range  of  temperatures  a  copper-iron  junction  in  an  electric  fur- 
nace (p.  107)  is  very  suitable.  If  the  current  that  heats  the 
furnace  is  steady,  the  furnace  will  heat  up  at  a  fairly  uniform 
rate,  and  a  curve  of  thermal  e.  m.  f.  against  time  will  show  quali- 
tatively the  way  in  which  the  thermal  e.  m.  f.  varies  with  the 
temperature.  If  a  platinum  resistance  thermometer  is  available, 
the  temperature  may  also  be  obtained  and  used  in  plotting  a 
curve. 

Questions. 

1.  State  what  would  be  observed  if  the  temperature  of  the  hot  junction 
were  increased  steadily  beyond  the  highest  temperature  used  in  this  experiment. 

2.  Is  the  effect  observed  here  due  solely  to  differences  of  potential  produced 
at  the  contacts? 

3.  For  what  range  would  each  couple  tested  be  suitable  as  a  thermometer? 
Explain. 


LXIX.    ELEMENTARY  STUDY  OF  RESISTANCE,  SELF- 
INDUCTION  AND  CAPACITY. 

References — Elementary:  Duff,  §523;  Hadley,  pp.  210,  441-450;  Kimball,  §§749- 
752;  Reed  &  Guthe,  §§347-348;  Watson,  §§451,  531. — More  Advanced: 
Parr's  Electrical  Engineering  Testing,  pp.  367-371. 

In  the  following  exercises,  which  are  intended  for  students 
who  have  not  made  a  study  of  the  theory  of  alternating  currents, 
some  of  the  properties  of  such  currents  are  studied  and  compared 
with  those  of  direct  currents. 

Ohm's  Law  for  steady  currents  states  that 

T? 

—  =  a  constant  =  R, 
i 

where  R  is  called  the  resistance  of  the  conductor. 

(A)  Apply  various  e.  m.  f.'s  to  a  non-inductive  conductor. 
Measure  the  current  by  an  ammeter,  and  the  voltage  by  a  volt- 
meter (of  any  type)  and  calculate  R  for  each  value  of  the  e.  m.  f. 
The  latter  may  be  varied  by  means  of  a  series  rheostat. 

(B)  Apply  the  same  method  to  (i)  a  large  coil,  (2)  the  large  coil 


214  ELECTRICITY   AND   MAGNETISM. 

and  the  non-inductive  resistance,  (a)  in  series,  and,  (b)  in  parallel. 
Compare  the  results  of  (a)  and  (b)  with  the  calculated  values. 

(C)  Repeat  (A)  and  (B),  using  alternating  currents  and  an 
electrostatic  voltmeter.     Corresponding  to  Ohm's  Law  we  have 

E  ____ 

-r  =  constant  =  \/R2-|-47r2n2L2, 

where  the  constant  is  called  the  impedance  and  L  is  the  coefficient 
of  self-induction  and  n  the  frequency.  Find  the  value  of  the 
constant  for  different  e.  m.  f.'s  and  currents,  and,  from  the  mean 
and  the  values  of  R  and  n,  calculate  L.  Contrast  the  results 

in  series  and  parallel  combinations 
with  the  values  calculated  by  treat- 
ing impedance  in  the  same  way  as 
resistance  in  direct  currents. 

Tabulate  all  results  so  that  they 
may  be  readily  compared. 

(D)  When  an  alternating  current  is 
applied  to  a  condenser,  it  is  charged, 
discharged,  charged  oppositely  and 
pIG  7g  discharged  during  each  alternation. 

Evidently  the  total  quantity  that 
traverses  the  leads  in  each  unit  of  time  is  proportional  to  the  fre- 
quency and  to  the  product  of  the  capacity  and  the  voltage  (since 
q  =  CV)  and  it  can  be  shown  that  the  current  is  given  by 


Measure  i  for  various  values  of  V  and  calculate  C.  Do  this 
for  several  condensers  (i)  separately,  (2)  in  parallel,  (3)  in  series. 
Compare  the  results  of  (2)  and  (3)  with  the  calculated  values. 

Questions. 

1.  What  is  meant  by  the  effective  value  of  an  alternating  current  and  what 
ratio  does  it  bear  to  the  maximum  value? 

2.  How,  by  means  of  a  diagram,  would  you  find  the  impedance  when  given 
the  ohmic  resistance  R  and  the  inductance  L? 

3.  Supposing  the  alternating  e.  m.  f.  resolved*  graphically  into  two  parts, 
one  to  overcome  the  ohmic  resistance  and  the  other  to  overcome  the  inductance, 
what  relation  between  the  phases  of  these  two  parts  does  question  (2)  suggest? 

4.  Explain  construction  and  peculiar  advantages  of  electrostatic  voltmeter. 


INDUCTION   AND   CAPACITY,    ALTERNATING   CURRENTS.       215 

LXX.    SELF-INDUCTION,  MUTUAL  INDUCTION  AND 
CAPACITY,  ALTERNATING  CURRENTS. 

See  references  to  LXIX.  Jackson's  Alt.  Cur.,  pp.  90-91,  151-200;  Parr's  Elec- 
trical Eng.  Testing,  pp.  222-224,  228-231,  234-235;  /.  /.  Thomson's  Elec- 
tricity and  Magnetism,  §§233,  244-245.  For  Electrostatic  Voltmeter,  see 
Parr,  367-371- 

This  exercise,  which  is  somewhat  more  advanced  than  the  pre- 
ceding, is  intended  for  students  who  have  made  some  study  of 
the  theory  of  alternating  currents. 

Let  E  be  the  alternating  e.  m.  f.  in  a  circuit  of  resistance  R, 
capacity  C,  and  self-induction  L.  If  i  is  the  current, 

E 


We  can  test  the  above  formula  by  calculation,  after  measuring 
i,  E,  R,  C,  and  L.  An  inductance  with  a  magnetic  core  has  a 
variable  value  of  L,  the  magnitude  of  which  depends  on  the 
strength  of  the  current.  Hence,  for  this  experiment,  an  in- 
ductance consisting  of  a  very  large  coil  containing  no  iron  is  used. 

(A)  Measurement  of  C. — If,  in  the  general  formula,  L  be  zero, 
C  can  be  deduced  from  the  values  of  i,  E,  and  R,  assuming  that  co, 
which  equals  2  TT  times  the  frequency  n,  is  known.     E,  the  e.  m. 
f .  across  the  terminals  of  the  condenser,  is  measured  by  an  electro- 
static voltmeter,  i  by  an  alternating  current  ammeter.     Initially 
a  high  resistance  of  large  current  capacity  must  be  included. 
This  may  later  be  cut  out.     A  fuse  of  lower  capacity  than  the 
range  of  the  ammeter  must  be  permanently  in  circuit. 

(B)  Measurement  of  L. — The  value  of  L  is  found  by  observing 
the  values  of  i  and  E  in  a  circuit  containing  the  self-inductance 
coil  and  then  applying  the  general  formula.     Sufficient  additional 
resistance  must  be  placed  in  the  circuit,  but  the  value  of  E  re- 
quired is  that  across  the  terminals  of  the  inductance  coil.     R, 
which  in  this  case  is  the  resistance  of  the  coil,  is  best  found  by 
Wheatstone's  Bridge 

(C)  Test  of  General  Formula. — Connect  the  condenser  and  self- 
induction  in  series.     Measure  the  current  and  the  total  e.  m.  f. ; 
also  the  e.  m.  f.  across  each  part.     Connect  the  condenser  and 


216  ELECTRICITY   AND   MAGNETISM. 

inductance  coil  in  parallel.  Measure  the  common  e.  m.  f.,  the 
total  current  and  the  current  in  each  branch. 

Calculate  i  in  the  series  arrangement  from  the  above  formula 
and  compare  with  the  experimental  value.  If  you  are  familiar 
with  the  method  of  complex  quantities  and  graphical  methods, 
apply  these  also  to  calculate  the  currents  in  both  series  and 
parallel  arrangements. 

(D)  Measurement  of  Mutual  Inductance.  —  Measure  the  mutual 
inductance,  M,  of  the  two  coils  of  a  transformer  (with  iron  core) 
by  observing  the  e.  m.  f.,  E,  across  one  coil  when  a  measured 
current,  i,  is  applied  to  the  other. 


Vary  i  several  times  and  find  how  M  varies. 

Questions. 

1.  Why  is  an  electrostatic  voltmeter  necessary? 

2.  Does  the  self-induction  depend  upon  the  frequency?     Why  does  the 
latter  enter  into  the  equation? 


LXXI.    DIELECTRIC  CONSTANTS  OF  LIQUIDS. 

References — Elementary:  Duff,  §413;  Ames,  pp.  641,  66 1;  Hadley,  Chap.  X; 
Spinney,  §262;  Watson,  §452. — More  Advanced:  Faraday's  Experimental 
Researches  in  Electricity  and  Magnetism,  §§1252-1306. 

The  dielectric  constant  of  a  liquid,  or  the  ratio  of  the  capacity 
of  a  condenser  with  that  liquid  as  dielectric  to  its  capacity  when 
its  dielectric  is  air,  can  be  determined  by  a  comparison  of  capaci- 
ties by  the  Bridge  Method  of  Exp.  LX.  For  this  purpose  it  is 
convenient  to  use  a  condenser  consisting  of  two  parallel  plates, 
the  distance  between  which  is  adjustable,  as  shown  in  Fig.  79. 
The  distance  between  the  plates  can  be  measured  by  means  of  a 
scale,  B,  attached  to  the  movable  plate  A,  and  a  vernier  attached 
to  the  framework.  The  plates  hang  in  a  vessel  for  holding  the 
dielectric.  Two  methods  can  be  used.  In  one  the  distance  be- 
tween the  plates  is  not  varied;  in  the  other  it  is  varied. 

The  first  method  consists  in  comparing  the  capacity  of  the 
above  condenser  with  that  of  a  Leyden  jar  (i)  when  the  dielectric 
is  the  liquid  to  be  tested;  (2)  when  it  is  air.  From  these  com- 


DIELECTRIC    CONSTANTS   OF   LIQUIDS. 


217 


parisons  the  ratio  of  the  capacities  of  the  condenser  in  the  two 
conditions  is  deduced  and  this  equals  the  dielectric  constant  of 
the  liquid.  Instead  of  a  battery  and  galvanometer,  an  induction 
coil  and  a  sensitive  telephone  are  used. 


£ 


A     A' 


FIG.  79. 
The  second  method  assumes  the  (approximate)  formula 


Ae 


c= 


for  the  capacity  of  such  a  plate  condenser  (in  electrostatic  units, 
see  p.  154),  where  A  is  the  area  of  each  plate,  e  is  the  dielectric 
constant  of  the  surrounding  medium,  and  d  is  their  distance  apart. 
If  C  be  the  same  with  two  dielectrics,  but  with  different  values 
ofd 


Having  obtained  a  balance  for  air  as  dielectric,  leave  the 
resistances  in  the  bridge  unchanged  and  again  obtain  a  balance, 
after  filling  the  jar  with  the  liquid  to  be  tested,  by  adjusting  the 
distance  between  the  plates.  This  second  method  is  less  accurate, 
since  the  formula  assumed  is  only  approximate  and  the  distances 
cannot  be  determined  as  accurately  as  the  resistances.  An 
accurate  formula  will  be  found  in  Kohlrausch's  Physical  Measure- 
ments, p.  379. 


2l8  ELECTRICITY   AND   MAGNETISM. 

The  above  methods  should  be  applied  to  two  highly  insulating 
liquids,  such  as  kerosene  and  benzol. 

Questions. 

1.  Calculate  the  capacity  of  the  two  plates  when  separated  by  (a)  air;  (b) 
liquid,  the  distance  apart  being  the  same  as  in  the  first  part  of  the  experiment. 

2.  Calculate  the  charge  for  each  case  if  (a)  100  electrostatic  units  of  potential 
are  applied;  (b)  100  volts  (p.  154). 


LXXII.   ELECTRIC  WAVES  ON  WIRES. 
Dielectric  Constants  of  Liquids. 

References—Elementary:  Duff,  §§541-542,  544-545;  Ames,  pp.  752-754;  Hadley, 
PP-  541-548,  555-556;  KMall,  §§762-767;  Reed  &  Guthe,  §544;  Spinney, 
§§384-390;  Watson,  §588.— More  Advanced:  Drude,  Ann.  der  Phys.,  Vol.  8, 
p.  336;  Kohlrausch's  Physical  Measurements  (on  capacity  of  a  plate  conden- 
ser), p.  379;  Poincare,  Maxwell's  Theory;  J.  J.  Thomson's  Elements  of 
Electricity  and  Magnetism,  §  243. 

In  this  experiment  electric  waves  on  a  wire,  AD,  are  excited 
by  electric  oscillations  in  a  neighboring  circuit  or  "exciter,"  E, 
which  contains  an  inductance,  L,  and  a  capacity,  C.  The  period 
of  such  oscillations  is  T  =  2*VZc7  The  inductance  is  that 
of  two  thick  semicircular  wires.  The  ends  e  and  e'  of  these  wires 
carry  small  spheres  and  the  wires  are  so  bent  that  the  spheres 
are  beneath  the  surface  of  kerosene  in  a  small  cup  and  form  a 
spark-gap.  The  length  of  this  spark-gap  can  be  adjusted  by 
means  of  a  micrometer  screw  attached  to  one  of  the  ebonite 
posts,  H  Hr,  on  which  the  semicircular  wires  are  supported. 
The  condenser  is  of  the  variable  form  shown  in  Fig.  77  and  is 
connected  between  the  other  two  ends  of  the  semicircular  wires. 
The  impulses  that  start  the  oscillations  in  the  exciter  are  produced 
by  a  Tesla  coil,  the  secondary  of  which  is  connected  across  the 
spark-gap  ee' ,  while  the  primary  of  the  Tesla  coil  is  connected 
through  another  spark-gap,  Z,  to  the  secondary  of  an  induction 
coil  /. 

The  wire,  AD,  on  which  the  waves  are  formed  is  bent  to  a 
Z7-form  and  lies  in  a  horizontal  plane  above  the  plane  of  the  ex- 
citer. The  oscillations  in  the  exciter  act  by  induction  on  the 
part,  A ,  of  the  wire  and  produce  waves  that  move  along  the  wire 
toward  D;  between  any  two  corresponding  points,  such  as  d 


ELECTRIC   WAVES   ON   WIRES. 


219 


and  d',  there  is  an  oscillating  difference  of  potential  and  the  trans- 
mission of  this  oscillation  constitutes 
the  wave-motion.  At  the  free  end,  D, 
these  waves  are  reflected  and  interfere 
with  the  direct  waves.  If  the  wire  is 
of  proper  length,  this  interference  pro- 
duces stationary  waves;  that  is,  the 
wire  "resonates"  to  the  exciter.  If 
the  wire  be  also  sufficiently  long,  there 
will  be  one  or  more  nodes  on  the  wire, 
that  is,  places  of  no  potential  differ- 
ence, with  intervening  antinodes,  or 
places  of  maximum  oscillating  poten- 
tial difference.  If  a  small  wire 
"bridge,"  B,  be  placed  across  the 
wire  at  a  node  it  will  not  interfere 
with  the  stationary  waves ;  but  it  will 
destroy  them  if  it  is  placed  at  any 
other  point.  (It  will  be  instructive 
to  compare  the  above  with  the  forma- 
tion of  stationary  sound  waves  in  a 
resonance  tube  such  as  that  of  Exp. 
XXX,  the  tuning-fork  being  the  ex- 
citer.) When  the  bridge  has  been 
placed  at  a  node  the  part  of  the  wire 
between  it  and  the  free  end,  D,  could 
be  changed  in  length  or  removed  with- 
out appreciably  diminishing  the  oscil- 
lations between  A  and  B,  (just  as  the 
lower  part  of  a  violin  string  that  is 
touched  by  the  finger  does  not  inter- 
fere with  the  vibrations  of  the  upper 
part.)  The  part  A  of  the  wire  is  (ap- 
proximately) a  node,  although  it  is 
the  part  where  the  oscillations  are 
excited.  (Compare  with  this  the  fact 
that  when  a  tuning-fork,  connected  to 
a  long  thread  as  in  Melde's  experiment,  throws  the  latter  into 
stationary  vibrations,  the  point  of  connection  is  a  node). 


B 


D 

FIG.  80. 


d' 


B 


220  ELECTRICITY   AND   MAGNETISM. 

Various  means  have  been  used  for  detecting  such  stationary 
vibrations.  The  simplest  is  a  "vacuum"  tube,  F,  containing 
helium  (or  neon)  at  a  very  low  pressure.  An  alternating  po- 
tential difference  between  the  ends  of  such  a  tube  will  cause 
oscillating  discharges  accompanied  by  a  glow.  If  placed  across 
the  wire  where  there  are  no  stationary  oscillations  the  tube  will 
not  glow;  but,  if  stationary  oscillations  exist,  it  will  glow  brightly 
at  an  antinode,  less  brightly  between  an  antinode  and  a  node, 
not  at  all  at  a  node.  By  the  aid  of  the  tube  the  bridge  can  be 
adjusted  to  each  node.  If  the  wire  be  long  enough  to  permit  of 
more  than  one  node,  twice  the  distance  between  two  adjacent 
nodes  will  equal  the  wave  length. 

Since  the  wire  is  in  resonance  with  the  exciter,  the  period  of 
oscillation  of  the  wire  equals  that  of  the  exciter.  The  latter, 
and  therefore  the  former,  can  be  changed  by  changing  C.  Hence 


The  electric  waves,  while  directed  by  the  wire,  are  really  waves 
of  oscillation  of  electric  force  in  the  medium  between  the  two 
branches  of  the  wire.  Such  waves  travel  with  a  velocity  that  is 
independent  of  the  wave  length,  and,  if  Xi,  be  the  wave-length 
when  the  period  is  TI,  X2  that  when  the  period  is  T2,  v  =  Xi/Tj. 
=  X2/T2.  Hence 

G=xL2 

€2     X22 

By  determining  the  wave-length  with  air  as  the  dielectric  in  C 
and  then  with  a  liquid  as  dielectric,  we  can  evidently  find  the 
dielectric  constant  of  the  liquid. 

In  the  practice  of  the  method  most  trouble  is  likely  to  be  due 
to  the  spark-gap,  eef.  It  must  be  adjusted  until  the  spark  occurs 
under  the  kerosene.  To  avoid  danger  to  the  tube  by  accidental 
dropping,  it  may  be  attached  loosely  to  the  wire  by  a  loop  of 
thread.  Each  node  should  be  determined  several  times.  The 
distance  between  the  plates  of  C  should  be  varied  four  or  five 
times  with  air  as  dielectric  and  a  curve  drawn  with  d  as  abscissa 
and  X  as  ordinate.  From  this  curve  and  the  value  of  X  for  each 


ELECTRIC    WAVES    ON   WIRES.  221 

liquid,  the  value  of  the  dielectric  constant  for  that  liquid  is  readily 
deduced. 

For  a  complete  proof  of  the  formula  works  on  electricity  and  magnetism 
must  be  consulted  (e.  g.,  J.  J.  Thomson's  Electricity  and  Magnetism,  §243). 
The  following  considerations  suggest  the  formula.  When  an  alternating  e.  m. 
f.,  E,  is  applied  to  a  circuit  containing  capacity,  self-inductance,  and  ohmic 
resistance  in  series, 


The  potentials  across  the  self-inductance  (including  the  resistance)  and  the 
capacity,  respectively,  are 

EL  =  I^r2+tfL*  and  Ec=^ 

Evidently  the  potentials  across  the  parts  of  the  system  may  be  greater  than  the 
total  e.  m.  f.  which  is  the  resultant  obtained  by  geometrical  (i.  e.,  vector)  addi- 
tion of  the  parts.  This  constitutes  resonance.  It  is  complete  when 


Substituting  foV  w  its  value  2ir/T,  we  get  T  =  2irVLC.  This,  then,  is  the 
period  of  the  applied  e.  m.  f.  when  resonance  results.  It  is  also,  therefore,  the 
period  of  the  free  natural  vibrations  of  the  system. 

The  approximate  formula  for  the  capacity  of  a  plate  condenser  is  given  in 
Exp.  LXII,  a  more  exact  one  in  Kohlrausch,  p.  379. 

Questions. 

1.  Assuming  the  velocity  of  the  waves  to  be  that  of   light,  calculate  the 
frequency  of  the  oscillations  for  one  value  of  X. 

2.  From  this  and  the  approximate  formula  for  C  calculate  L. 

3.  If  a  sufficient  amount  of  liquid  were  available,  how  could  its  dielectric 
constant  be  found  by  immersing  the  wire  AD  in  it? 

4.  What  would  be  observed  if  AD  were  contained  in  a  vacuum  tube? 


TABLES. 


In  many  of  these  tables  the  data  are  the  means  of  the  most  re- 
liable data  obtained  by  different  observers.  When  more  com- 
plete information  is  required  the  student  should  refer  to  Landolt 
and  Bernstein's  Tabellen. 


224 


TABLES. 


TABLE  I. 

Logarithms  of  Numbers  from  I  to   1000. 


No. 

0 

I 

2 

3 

4  |  5 

6 

7   |  8 

9 

10 

0000 

0043 

0086 

0128 

0170 

O2I2 

0253 

0294 

°334 

°374 

ii 

0414 

04^3 

0492 

Q531 

0569 

0607 

0645 

0682 

0719 

°755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1  106 

13 

1139 

11  73 

I2O6 

1239 

1271 

J303 

1335 

1367 

1399 

J43° 

14 

1461 

1492 

1523 

*553 

1584 

1614 

1644 

1673 

1703 

J732 

15 

1761 

1790 

1818 

1847 

I875 

1903 

J93i 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2i75 

22OI 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

262  5 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3°54 

3°75 

3096 

3118 

3*39 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

33°4 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

354i 

356° 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

43*4 

433° 

4346 

4362 

4378 

4393 

4409 

4425 

444° 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

47X3 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997  ' 

501  1 

5024 

5038 

32 

5051 

5°65 

5079 

5092 

5I05 

5JI9 

5J32 

5*45 

5*59 

5J72 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

53i5 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

54i6 

5428 

35 

544i 

5453 

5465 

5478 

549° 

5502 

55J5 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5  7°  5 

57J7 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

591* 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284. 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

65°3 

65*3 

6522 

45 

6  S3  2 

6542 

6551 

6-561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7°33 

7042 

7050 

7°59 

7067 

5i 

7076 

7084 

7°93 

7101 

7110 

7118 

7126 

7J35 

7M3 

7J52 

52 

7160 

7168 

7i77 

7185 

7J93 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

73i6 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

738o 

7388 

7396 

No. 

o 

i 

2 

3 

4  1  S 

6 

7 

8 

9 

TABLES. 


225 


TABLE  I.— Continued. 
Logarithms  of  Numbers  from  i  to  1000. 


No. 


55 

7404 

7412 

7419 

7427 

7435 

7443 

745i 

7459 

7466 

7474 

56 

7482 

7490 

7497 

75°5 

75i3 

7520 

7528 

7536 

7543 

755i 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

773i 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8i95 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

7i 

8513 

8519 

8525 

853i 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

859i 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

875i 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9<?i5 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9J33 

82 

9138 

9i43 

9149 

9J54 

9J59 

9165 

9170 

9i75 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

93°4 

93°9 

93  1  5 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

937° 

9375 

9380 

9385 

939° 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

945° 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

95*3 

95i8 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

958i 

9586 

9i 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

97°3 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

974i 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

No. 

0 

I 

2 

3 

4 

5 

6 

7 

8  |  9< 

226 


TABLES. 


TABLE  II. 
Natural  Sines  and  Cosines. 


Sine 

57o  || 

Cosine 

Di° 

^  0 

o.oooo 

90 

I.OOOO 

I 

0.0175 

175 

89 

0.9998 

02 

2 

0.0349 

*74 

88 

0.9994 

04 

3 

0.0523 

J74 

87 

0.9986 

08 

4 

0.0698 

175 

86 

0.9976 

10 

5 

0.0872 

174  - 

85 

0.9962 

14 

6 

0.1045 

173 

84 

0.9945 

*7 

7 

0.1219 

174 

83 

0.9925 

20 

8 

0.1392 

173 

82 

0.9903 

22 

9 

0.1564 

172 

81 

0.9877 

26 

10 

0.1736 

172 

80 

0.9848 

29 

ii 

0.1908 

172 

79 

0.9816 

32 

12 

0.2079 

171 

78 

0.9781 

35 

13 

0.2250 

171 

77 

0-9744 

37 

14 

0.2419 

169 

76 

0.9703 

4i 

15 

0.2588 

169 

75 

0.9659 

44 

16 

0.2756 

168 

74 

0.9613 

46 

i? 

0.2924 

168 

73 

0-9563 

5o 

18 

0.3090 

166 

72 

0.9511 

52 

iQ 

0.3256 

1  66 

7i 

0-9455 

56 

20 

0.3420 

164 

70 

0-9397 

58 

21 

0.3584 

164 

69 

0-9336 

61 

22 

0.3746 

162 

68 

0.9272 

64 

23 

0.3907 

161 

67 

0.9205 

67 

24 

0.4067 

1  60 

66 

0-9135 

70 

25 

0.4226 

*59 

65 

0.9063 

72 

26 

0.4384 

158 

64 

0.8988 

75 

27 

0.4540 

156 

63 

0.8910 

28 

0.4695 

155 

62 

0.8829 

81 

29 

0.4848 

153 

61 

0.8746 

83 

3° 

0.5000 

152 

60 

0.8660 

86 

31 

0.5150 

T-50 

59 

0.8572 

88 

32 

0.5299 

149 

58 

0.8480 

92 

33 

0.5446 

147 

57 

0.8387 

93 

34 

0.5592 

146 

56 

0.8290 

97 

35 

0.5736 

144 

55 

0.8192 

98 

36 

0.5878 

142 

54 

0.8090 

IO2 

37 

0.6018 

140 

53 

0.7986 

104 

38 

0.6157 

139 

52 

0.7880 

106 

39 

0.6293 

136 

51 

0.7771 

109 

40 

0.6428 

J35 

5° 

0.7660 

in 

4i 

9.6561 

J33 

49 

0-7547 

"3 

42 

0.6691 

130 

48 

0.7431 

116 

43 

0.6820 

129 

47 

0.73*4 

117 

44 

0.6947 

127 

46 

o.7i93 

121 

45 

0.7071 

124 

45o  t 

0.7071 

122 

Cosine 

D  i°  || 

Sine 

Dl° 

TABLES. 


227 


TABLE  III. 
For  Reduction  of  Time  of  Oscillation  to  an  Infinitely  Small  Arc. 

i          ^      5  a 

k=—  sin2  -T+ZT  sin4  — . 
4          4     04          4 

If  /  =  observed  time  and 
T  =  true  reduced  time 
T  =  t-kt. 


a 

k 

a 

k 

a 

k 

0° 

o.ooooo 

7° 

O.00023 

H° 

0.00093 

i 

ooo 

8 

030 

15 

107 

2 

002 

9 

039 

16 

122 

3 

004 

10 

048 

17 

138 

4 

008 

ii 

058 

18 

154 

5 

012 

12 

069 

19 

172 

6 

017 

13 

080  . 

20° 

190 

7° 

023 

14° 

093 

TABLE  IV. 
Reduction  of  Barometer  Readings  to  o. 

(The  corrections  below  are  in  mm.  and  are  to  be  subtracted.     The  uncor- 
rected  height  is  in  cm.) 


Temp. 

Brass  Scale 

Glass  Scale 

72 

73 

74 

75 

76 

77 

78 

74 

75 

76 

77 

78 

i5 

*-7S 

1.77 

1.81 

1.83 

1.86 

1.88 

1.91 

1.92 

1.94 

1.97 

2.OO 

2.  02 

1  6 

1.87 

1.89 

i-93 

1.96 

1.98 

2.OI 

2.03 

2.05 

2.07 

2.IO 

2.13 

2.16 
2  29 

I7 

1.98 

2.OI 

2.05 

2.08 

2.IO 

2.13 

2.16 

2.17 

2.  2O 

2.23 

2.26 

18 

2.IO 

2.13 

2.17 

2.  2O 

2.23 

2.26 

2.29 

2.30 

2.33J2.36 

2-39 

2-43 

J9 

2.22 

2.25 

2.29 

2.32 

2-35 

2.38 

2.41 

2-43 

2.46 

2.49 

2-53 

2.56 

20 

2-33 

2-37 

2.41 

2.44 

2.47 

2-51 

2.54 

2.56 

2-59 

2.62 

2.66 

2.69 
2.83 
2.96 

2  I 

2.45 
2-57 

2.48 

2-53 
2.65 

2.56 
2.69 

2.60 
2.72 

2.63 

2.67 

2.68 

2.72 

2.76 

2.79 

22 

2.60 

2.76 

2.79 

2.81 

2-85 

2.89 

2.92 

23 

2.68 

2.72 

2,77 

2.81 

2.84 

2.88 

2.92 

2.94 

2.98 

3-02 

3.06 

3.10 

3-23 

24- 

2.80 

2.84 

2.89 

2.93 

2.97 

3-01 

3-05 

3.06 

3-n 

3-15 

3-i9 

25 

2.92 

2.96 

3.01 

3-°5 

3-°9 

3-J3 

3-J7 

3.19 

3-23 

3.28 

3-32 

3.36 

228  TABLES. 

TABLE  V. 
Density  and  Volume  of  One  Gram  of  Water  at  Different  Temperatures. 


Temp. 

Density 

Vol.  of  i.  gr. 

Temp. 

Density 

Vol.  of  i.  gr. 

0° 

0.999878 

I.  000122 

21° 

0.998065 

.001939 

i 

Q-999933 

1.000067 

22 

0.997849 

.0021  56 

2 

0.999972 

1.000028 

23 

0.997623 

.002383 

3 

0.999993 

1.000007 

24 

0.997386 

.002621 

4 

I.OOOOOO 

I.OOOOOO 

25 

0.997140 

.002868 

5 

0.999992 

1.000008 

30 

0-99577 

.00425 

6 

0.999969 

1.000031 

35 

0.99417 

.00586 

7 

0-999933 

1.000067 

40 

0.99236 

.00770 

8 

0.999882 

•  i.oooi  18 

45 

0.99035 

.00974 

9 

0.999819 

1.  000181 

5° 

0.98817 

.01197 

10 

0-999739 

1.000261 

55 

0.98584 

.01436 

1  1 

o  999650 

1,0003  5° 

60 

0-98334 

.01694 

12 

0.999544 

1.000456 

65 

0.98071 

.01967 

13 

0.999430 

1.000570 

70 

0.97789 

.O226l 

14 

0.999297 

1.000703 

75 

0-97493 

.02570 

15 

0.999154 

1.000847 

80 

0.97190 

.02891 

16 

0.999004 

1.000997 

85 

0.96876 

.03225 

J7 

0.998839 

i.ooi  162 

90 

0.96549 

•03574 

18 

0.998663 

1.001339 

95 

0.96208 

.03941 

J9 

0.998475 

1.001527 

IOO 

0.95856 

•04323 

20 

0.998272 

1.001731 

TABLE  VI. 

Density  of  Gases  (o°,  76  cm.).1 

Hydrogen 00008987 

Oxygen .0014290 

Nitrogen 0012507 

Air    0012928 

Chlorine 003167 

Carbon  monoxide 0012504 

Carbon  dioxide 0019768 

Ethane 001341 

Ethylene 001252 

Steam  (at  100°) 00060315 

J  Largely  from  Guye,  J.  Ch.  Phys.,  1907,  p.  203. 


TABLES.  229 

TABLE  VII. 

Density  (o°),  Specific  Heat  (o°),  and  Coefficient  of  Linear  Expansion. 


Element 

Density 

Specific 
Heat 

Coef.  of  Lin.  Exp. 
Multiplied  by  10°. 

Aluminum 

2.60 

.21 

2^.1 

Antimony 

6.62 

.O4.Q 

Bismuth 

9.8 

.O1>I 

Cadmium    . 

8.61 

.OSS 

T.o.7 

Carbon,  diamond  
Carbon,  graphite  
Carbon,  gas  carbon  .... 
Cobalt  

3-52 
2.25 
i.  90 
8.8 

.10 
•15 

I  O6 

3i.i8 

7-8 

54 
12.4 

Copper 

8.Q2 

.00  s 

168 

Copper  sulphate  (crys.)  . 
Gold        .      . 

3.58 
IQ.7 

.0^2 

14.4 

Iron 

7.8 

.11 

12.  1 

Lead  .... 

n.^6 

.0^1 

29.2 

Magnesium  
Mercury  
Nickel  
Phosphorus,  yellow  .... 
Phosphorus,  red  

1.74 
13.596 
8.9 
1.83 
2.19 

.0333 
.108 
.20 
.17 

181  (cub.  exp.) 

12.8 

Phosphorus,  metallic  .  .  . 
Platinum 

2-34 

21.4. 

.QT.-1 

9.0 

Potassium  chloride  
Silver 

1.98 

io.s^. 

.os6 

19.2 

Sodium  chloride    . 

2.  IS 

Sodium  sulphate  
Tin    

2.65 
7.7 

.0^6 

22.^ 

Zinc   . 

7.2 

.004. 

29.2 

Zinc  sulphate  (anhy.)  .  .  . 

349 

230 


TABLES. 
TABLE  VIII. 


Density,  Specific  Heat,  and  Coefficient  of  Expansion  of  Miscellaneous 

Substances  (o°) 


Substance 

Density 

Specific 
heat 

Coef.  of  Lin.  Exp. 
(Xio°) 

Castor  oil  
Glass,  green  
Glass,  crown  

.969 

2.6 

2.7 

.19 
.20 

8.9 
88 

Glass,  crystal  

2.9 

.18 

7.7 

Glass  flint 

T.,1  C—  -J   Q 

IQ 

7  <j 

Hard  rubber  
Marble  .  .  . 

I-I5 

2.75 

7-7 

117 

Paraffin  

.89 

Quartz,  crystal  1  1  
Quartz,  crystal  _|_  
Quartz,  fused  
Alcohol  (ethyl)  

2.653 
2.653 
2.20 

.81 

.19 

•  54 

7.2 
13.2 

•54 

I  0481 

Benzol  

.899 

.38 

i  I761 

Carbon  bisulphide  
Chloroform 

1.293 

I     C-2 

.24 
27 

I.I41 

i  ii1 

Ether  (ethyl)  
Glycerine    .  . 

•74 
1.26 

•53 
58 

i-Si1 

TABLE  IX. 
Average  Value  of  Elastic  Moduli. 


I 

Shear  Modulus. 

Young's  Modulus. 

Brass    .... 

t.  7  X  lo11 

10  4  X  ion 

Iron  

7  7  X  ion 

IQ  6  X  lo11 

Steel  

8.2  X  lo11 

22        X   IO11 

1  Coefficient  of  cubical  expansion  X  io3. 


TABLES.  231 

TABLE  X. 

Surface  Tension  T  (15°),  Temperature  Coefficient  of  Surface 
Tension  c',  and  Angle  of  Contact  a. 


T 

c' 

a 

Ethyl  ether    

I  0 

I  I 

1  6° 

Ethyl  alcohol  

25 

-.087 

0° 

Benzol 

•?  i 

I  3 

0° 

Water  

76 

—  i  c 

f  small 

/u 

Mercury 

<2  7 

—  18 

IT.  s° 

TABLE  XI. 
Coefficient  of  Viscosity  (2o°).i 

Water 

Mercury 

Acetic  acid 

Methyl  alcohol 

Ethyl  alcohol , 

Ethyl  ether 

Benzol 


.0100 
.0159 

.0122 

.00591 

.0119 

.00234 

.00649 


TABLE  XII. 
Specific  Heats  of  Gases.2 


• 

Temp. 

Sp 

r=JLP 

'          SV 

Argon              

20° 
20° 

2750-3560 

0°-200° 
-30°-200° 
0°-200° 
0°-200° 

I9o"343o 
200°  -377° 

8s0-228° 

I30°-2500 
I0°-200° 
100° 
20°-2IO° 
28°-I  18° 
IIO°-220° 
70°-225° 

ii6°-2i8° 

.1205 

J-25 
.0246 

3.406 
.244 
.217 

.2375 

•  "5 
.0336 

•0555 
.480 

.245 
.217 
•512 
.144 

•453 
.480 

•  375 

1.66 
1.64 
1.66 
1.396 
1.405 
1.40 
1.405 
1.32 
1.29 
1.29 
1.287 
1.28 
1.28 
I-31? 
i-i54 
1.14 
i  .  -  . 

T      T»T 

Helium      

Mercury  

Hydrogen   

Nitrogen  

Oxygen  

Air 

Chlorine  '.  

Iodine 

Bromine 

Water 

Hydrogen   sulphide  .  .  . 
Carbon  dioxide    

Ammonia  

Chldroform  

Ethyl  alcohol    

Ether  

Benzol.  .  .  . 

1  Winkelmann,  1908,  I,  2,  p.  1397. 

2  Juptner,  Phys.  Chem.  I,  pp.  7i~73- 


232  TABLES. 

TABLE  XIII. 

Pressure  of  Saturated  Water  Vapor  (Regnault), 
(mm.) 


Temp. 

Pressure 

Temp. 

Pressure 

Ice                   Water 

29° 

29.782 

30 

3J-548 

—  10 

1.999                  2.078 

31 

33-405 

8 

2.379                  2.456 

32 
33 

3  5  3  59 
37-4io 

34 

39  565 

6 

2.821                   2.890 

35 

41.827 

40 

54.906 

4 

3-334                  3-387 

45 

7I-39I 

2 

3-925                  3-955 

5° 

55 

O  I  .  ()  o  2 

117.479 

60 

148.791 

65 

186.945 

0 

4.600 

7° 

233-093 

+     I 

4.940 

75 

288.517 

2 

5-302 

80 

354-643 

3 

5.687 

85 

433-41 

4 

6.097 

90 

525-45 

5 

6-534 

91 

545-78 

6 

6.998 

92 

566.76 

7 

7.492 

93 

588.41 

8 

8.017 

94 

610.74 

9 

8-574 

95 

633-78 

10 

9.165 

96 

657-54 

ii 

9.792 

97     • 

682.03 

12 

10.457 

98 

707.26 

J3 

11.062 

98-5 

720.15 

14 

11.906 

99-o 

733-91 

15 

12.699 

99-5 

746.50 

16 

13-635 

IOO.O 

760.00 

17 

14.421 

100.5 

773-71 

18 

15-357 

IOI.O 

787-63 

19 

16.346 

IO2.O 

816.17 

20 

I7-39I 

IO4.O 

875.69 

21 

18.495 

105 

906.41 

22 

19.659 

I  10 

1075.4 

23 

20.888 

I2O 

i49J-3 

24 

22.184 

I30 

2030.3 

25 

23-550 

J5o 

358i-2 

26 

24.998 

J75 

6717 

27 

26.505 

200 

1  1690 

28 

28.101 

225 

19097 

TABLES. 

TABLE  XIV. 
Boiling  Point  of  Water,  t,  at  Barometric  Pressure  p,  (mm.). 


233 


P' 

t. 

P- 

t. 

P. 

£. 

740 

99.26° 

75o 

99-630 

760 

100.00° 

41 

.29 

Si 

•67 

61 

.04 

42 

•33 

52 

•7° 

62 

.07 

43 

•37 

53 

-74 

63 

.11 

44 

.41 

54 

.78 

64 

.15 

45 

•44 

55 

.82 

65 

.18 

46 

.48 

56 

•85 

66 

.22 

47 

•S2 

57 

•89 

67 

.26 

48 

•56 

58 

•93 

68 

.29 

49 

•59 

59 

.96 

69 

•33 

750 

99-630 

760 

100.00° 

77° 

100.36° 

234 


TABLES. 


TABLE  XV. 

Wet  and  Dry  Bulb  Hygrometer. 

(Actual  vapor  pressures  (mm.)  for  different  temperatures  of  dry 
thermometer  and  various  differences  of  temperature  between  the  two 
thermometers. 

The  first  vertical  column  gives  the  temperature  of  the  dry-bulb 
thermometer.  The  first  horizontal  line  gives  the  difference  between 
the  two  thermometers.  Since  the  difference  is  zero  if  the  air  is  satu- 
rated, the  second  vertical  column  gives  the  saturated  vapor  pressure 
for  the  corresponding  temperatures  in  the  first  column.) 


t°c. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

I  I 

0 

4.6 

3-7 

2.9 

2.1 

i-3 

i 

4-9 

4.0 

3-2 

2.4 

1.6 

0.8 

2 

5-3 

4-4 

3-4 

2-7 

1.9 

I.O 

3 

5-7 

4-7 

3-7 

2.8 

2.2 

i-3 

4 

6.1 

5-1 

4.1 

3-2 

2.4 

1.6 

0.8 

5 

6-5 

5-5 

4-5 

3-5 

2.6 

1.8 

I.O 

6 

7.0 

5-9 

4-9 

3-9 

2-9 

2.O 

i.i 

7 

7-5 

6.4 

5-3 

4-3 

3-3 

2-3 

1,4 

0.4 

8 

8.0 

6.9 

5-8 

4-7 

3-7 

2-7 

1-7 

0.8 

9 

8.6 

7-4 

6-3 

5-2 

4.1 

3-1 

2.1 

i.i 

0.2 

10 

9.2 

8.0 

6.8 

5-7 

4.6 

3-5 

2-5 

!-5 

o-5 

ii 

9.8 

8.6 

7-4 

6.2 

5-1 

4.0 

2.9 

1.9 

09 

12 

10.5 

9.2 

8.0 

6.8 

5-6 

4-5 

3-4 

2-3 

!-3 

13 

I  1.2 

9.8 

8.6 

7-3 

6.2 

5-o 

3-9 

2.8 

*-7 

14 

II-9 

10.6 

9.2 

8.0 

6.7 

56 

4-4 

3-3 

2.2 

i.i 

15 

I2.7 

"•3 

9-9 

8.6 

7-4 

6.1 

5-o 

3-8 

2-7 

1.6 

o-5 

16 

13-5 

12.  I 

10.7 

9-3 

8.0 

6.8 

5-5 

4-3 

3-2 

2.1 

I.O 

J7 

14.4 

13.0 

"•$ 

IO.I 

8.7 

7-4 

6.2 

4.9 

3-7 

2.6 

*-$ 

0.4 

18 

15-4 

13-8 

12.3 

10.9 

9-5 

8.1 

6.8 

5-5 

4-3 

3-i 

2.O 

O.Q 

J9 

16.4 

14.7 

13.2 

11.7 

10.3 

8.9 

7-5 

6.2 

4-9 

3-7 

2-5 

1.4 

20 

17.4 

I5-7 

14.1 

12.6 

i  i.i 

9-7 

8-3 

.6-9 

5-6 

4-3 

3-1 

1.9 

21 

I8.5 

16.8 

^S-1 

13-5 

12.0 

10.5 

9.0 

7.6 

6-3 

5-° 

3-7 

2-5 

22 

19.7 

17.9 

16.2 

14.5 

I2.9 

11.4 

9-9 

8.4 

7-° 

5-7 

4-4 

3-J 

23 

20-9 

19.0 

i7-3 

15.6 

13-9 

12.3 

10.8 

9.2 

7-8 

6.4 

5-J 

3-8 

24 

22.2 

20.3 

18.4 

16.6 

14.9 

*3-3 

11.7 

IO.I 

8.7 

7-2 

5-8 

4-5 

25 

23-6 

21.6 

19.7 

17.8 

16.0 

14-3 

12.7 

i  i.i 

9-5 

8.0 

6.6 

S-2 

26 

25.0 

22.9 

21.0 

19.0 

17.2 

i5-4 

13-7 

12.  I 

10.5 

8.9 

7-4 

6.0 

27 

26.5 

24.9 

22.3 

20.3 

18.4 

16.6 

14.8 

I3-1 

11.4 

9.8 

8-3 

6.8 

28 

28.1 

25-9 

23-7 

21.7 

19.7 

17.6 

16.0 

14.2 

12.5 

10.8 

9.2 

7-7 

29 

29.8 

27-5 

25-3 

23.1 

2  I.I 

19.1 

17.2 

!S-3 

13.6 

11.9 

IO.2 

8.6 

3° 

31.6 

29.2 

26.9 

24.6 

22.5 

20.5 

18.5 

16.6 

14.7 

13.0 

I  1.2 

9.6 

TABLES. 

TABLE  XVI. 
Vapor  Pressure  of  Mercury  (mm.). 


235 


Temp. 

Pres. 

Temp. 

Pres. 

o 

O.£>2 

170 

8.091 

+  20 

O.04 

180 

I  1.000 

40 

0.08 

190 

14.84 

60 

0.16 

200 

19.90 

80 

o-3S 

210 

26.35 

100 

0.746 

220 

34  7° 

no 

1.073 

230 

45-35 

I2O 

1-534 

24O 

58.82 

130 

2-175 

250 

75-75 

I4O 

3-059 

26O 

96.73 

150 

4.266 

27O 

123.01 

I  60 

5.900 

.    280 

I55-I7 

TABLE  XVII. 

Melting  Point  of  Metals.     (Holborn  and  Day  and 
Waidner  and  Burgess.1) 


Tin 

Cadmium 
Lead.. .. 
Zinc  .... 
Antimony 


232 
321 

327 
419 

631 


Aluminum 657 

Silver , 961 

Gold Io642 

Copper Iog4 

Platinum *753r 


•  /oj 

1  Phys.  Rev.,  1909,  p.  467;  Compt.  Rend.,  1909,  cxlviii,  p. 
"    2  Carnegie  Inst.  Geophys.  Lab.,  1911,  p.  I. 


1177. 


236  TABLES. 

TABLE  XVIII. 
Wave  Lengths  in  Angstrom  Units  (io-8  cm.). 


Line 

Element 

Wave  Length 

Color 

C  H~. 

Hydrogen 

6  ^63  o  ^4 

Red 

DT 

Sodium 

^806  i  ?  £ 

Yellow 

D2  

Sodium 

5890  182 

Yellow 

F  H/9  .. 

Hydrogen 

4.86  1    ^27 

Blue 

G'  H,... 

Hydrogen 

A.T.A.O.  634 

Violet 

H      r     

Calcium 

^068   62  t? 

Violet 

Helium 

7o6  ^  2 

Red 

Helium 

6678.1 

Red 

Helium 

c;87  C  6 

Yellow 

Helium 

^Ol  <-7 

Green 

Helium 

402  I.O 

Blue 

Helium 

471'?   2 

Blue 

Helium 

447  I    ^ 

Violet 

Mercury 

6232  o 

Red 

Mercury 

C7QO  7 

Yellow 

Mercury 

55760.  6 

Yellow 

Mercury 

^460  7 

Green 

Mercury 

40  ^0-7 

Green—  Blue 

Mercury 

40  1  6.4 

Blue 

Mercury 

43  ^8.3 

Blue 

Mercury 

4078.1 

Violet 

Mercury 

4046.8 

Violet 

K« 

Potassium 

7600.3 

Red 

K/9. 

Potassium 

=5832.2 

Yellow 

Kf 

Potassium 

4047.4 

Violet 

rr 

L/isv 

Lithium 

6708  2 

Red 

Li/9    . 

Lithium 

6103.8 

Orange 

•^^ 

Cadmium 

6438  <; 

Red 

Cadmium 

<;o8q.8 

Green 

Cadmium 

47OO.O 

Blue 

TABLES. 


237 


TABLE  XIX. 

Refractive  Indices. 

[Yellow  light  (D  lines) ;  20°.] 

Glass,  light  crown  (density  =  2.50) 1.5280 

Glass,  heavy  crown  (density  =  3.00) 1.5604 

Glass,  light    flint       (density  =  2.87) 1.5410 

Glass,    heavy   flint  (density  =  4.22) 1.7102 

Quartz,  crystal,  J_,  ord 1-5442 

Quartz,  crystal,  1,  ext 1-5533 

Alcohol,  ethyl 1.3614 

Benzol 1.5014 

Carbon  bisulphide 1 .6277 

Chloroform 1.4490 

Ether,  ethyl l-356o 

Glycerine 1.4729 

Water 1.3329 

Air,  o°,  76  cm 1.000293 


TABLE  XX. 
Specific  Rotatory  Power  (20°).     Yellow  Sodium  Light  (D), 


Active  Substance 

Concentration 
(=c)  (gr.in  looc.c.) 

[#J° 

Cane-sugar   R 

(  nil 

66.639  —  .O2O8<7 

Invert  sugar  L   

\  10-86 
1—14. 

66.453  —  .OOOI24C 
2O.O7      —.O4IC 

Glucose    (dextrose).    R 
(crystallized)  
Fructose  (levulose)  L.  . 
Milk-sugar    R  

0-100% 
0-40 
e—  7 

47.73    +.oi5X% 
—  100.3      +  .io8<7 
52.53 

Tartaric  acid  R  

(  '51S 

15.06    -.131(7 

Quartz    R  or  L 

122-63 

13.  43  6  -.no<7 
2  1.  70  (for  i  mm.  thickness) 

Landolt  and  Bronstein. 


TABLES. 


TABLE  XXI. 
Photometric  Table. 
For  a  3QO-part  Photometric  Bar 


(300-w)2' 


n 

0 

i 

2 

3 

4 

50 

O.O4OO 

0.420 

0.0440 

0.0460 

0.0482 

60 

.0625 

-651 

.0678 

.0706 

•0735 

70 

.0926 

.0961 

.0997 

.1034 

.1072 

80 

.1322 

.1368 

.I4H 

.1643 

.1512 

90 

.1837 

.1896 

•1957 

.2018 

.2082 

100 

.2500 

.2576 

.2653 

•2734 

.2815 

no 

•3352 

•3449 

•3549 

-3652 

•3756 

120 

•4445 

•4570 

.4608 

.4829 

.4964 

130 

.5848 

.6009 

.6173 

•6343 

.6516 

140 

.7656 

.7864 

.8078 

.8296 

.8521 

150 

I.OOOO 

1.027 

1-055 

1.083 

1.113 

1  60 

1.306 

1-342 

1-379 

1.416 

1-454 

170 

1.700 

1-757 

i.  806 

1-856 

1.907 

1  80 

2.250 

2.313 

2-379 

2.446 

2.516 

190 

2.983 

3.070 

3.160 

3-253 

3-359 

2OO 

4.000 

4.122 

4.285 

4.380 

4.516 

210 

5-444 

5.621 

5-803 

5-994 

6.192 

220 

7-563 

7.826 

8.100 

8.387 

8.687 

230 

10.80 

II.  21 

11.64 

12.09 

12.57 

240 

16.00 

16.68 

17.41 

18.17 

18.98 

250 

25.00 

n 

5 

6 

7 

8 

9 

50 

0.0504 

0.0527 

0.0550 

0.0574 

0.0599 

60 

.0765 

.0796 

.0827 

.0859 

.0892 

70 

.mi 

.1151 

•1193 

•1235 

.1278 

80 

•1563 

.1615 

.1668 

.1723 

.1779 

90 

.2148 

.2215 

.2283 

•2354 

.2428 

100 

.2899 

.2985 

.3074 

.3164 

•3256 

no 

•3864 

•3974 

.4088 

.4204 

•4323 

120 

.5012 

•5244 

-5389 

•5538 

.5691 

130 

.6694 

-6877 

.7064 

.7257 

•7454 

140 

.8752 

.8988 

.9231 

.9481 

•9737 

150 

I.I43 

1.174 

1.205 

1.238 

1.272 

1  60 

1.494 

1-535 

1-577 

1.620 

1.664 

170 

1.960 

2.014 

2.071 

2.120 

2.188 

180 

2.588 

2.662 

2-739 

2.817 

2.889 

190 

3-449 

3-552 

3-658 

3-768 

3.882 

200 

4.656 

4.803 

4-954 

5.III 

5-275 

2IO 

6.398 

6.122 

6-835 

7.068 

7.310 

220 

9.000 

9-327 

9.670 

10.03 

10.40 

230 

13.07 

13.60 

14-15 

14.74 

15-35 

240 

19.84 

20.75 

21.72 

22.75 

23.84 

250 

TABLES. 


239 


TABLE  XXII. 
Specific  Resistance  at  o°  C.  and  Temperature  Coefficient. 


Specific 
Resistance 

Temperature 
Coefficient 

Bismuth  (hard)  
Copper  (annealed)  

132.6  X  io-6 
I.59O  X  io-6 

0.0054 
0.0043 

Copper  (hard  drawn)  .  .  .'  

1.622  X  io-6 

German  silver  (4.Cu  -f  2Ni  +  iZn) 
Iron 

20.24  X  io-6 
10.43  X  io-6 

0.00027 
0.007 

Lead  (pressed) 

19.85  X  io-6 

0.0039 

Mercury  

94.07  X  io-6 

0.00089 

Platinum  

8.957  X  io-6 

O.OO'?4 

Silver  (annealed)  

1.521  X  io-6 

0.00377 

Silver  (hard  drawn)  

1.652  X  lo-6" 

Tin 

0.565  X  io-6 

O.OO4. 

TABLE  XXIII. 
Specific  Resistance  and  Temperature  Coefficient  of  Solutions  (18°).* 


Sp.  Res. 

Temp. 
Coef. 

Sp.  Res. 

Temp. 
Coef. 

wHCl  .  . 

•J.72 

0.0165 

wNaCl  

11.4.5 

0.0226 

o.iwHCl  .  .  .  .  

28.5 

o.iwNaCl  

1  08.  i 

o.oiwHCl  
wHNO3 

271. 
327 

o  163 

o.oiwNaCl  
wKCl 

974- 
io  18 

o  02  1  7 

o.i«HNO3  

28.6 

o.iwKCl  

89.  5 

o.oi«HNO3  

272. 

o.oiwKCl  

817. 

w£H2SO4  

c  OS 

0.0164 

nAgNOs 

14.  75 

o  0216 

o.iw£H2SO4.  .  . 

4-4..  4- 

o.iwAgNOa 

IO5-7 

o.oiw£H2SO4  
«C2H4O2  

325- 

758. 

o.oiwAgNOa  .... 

w£Pb(NO3)2.  . 

922. 
23..  8 

o.iwC2H4O2  
o.omC2H4O2  

wNaOH 

2170. 
6990. 

6  25 

o  019 

o.iw^Pb(N03),  .  . 
o.oiw£Pb(NO3)2  . 

w^ZnSO4 

129.4 
967. 

T>7  6 

O  O25 

o.iwNaOH 

C4.  7 

o  i«|ZnSO4 

217 

o.oiwNaOH  

«NH4OH.. 
o.iwNH4OH  

500. 
1125. 

^CMO. 

.  . 

o.oiw^ZnSO4.  .  .  . 

niCuSO4  
o.iw^CuSO4 

1362. 

38.8 
22^. 

0.0225 

o.oiwNH4OH..  .  . 

10420. 

O.OI«5CuSO4.  .  .  . 

1385. 

•• 

*A  normal  solution  (designated  by  the  prefix  »),  contains  in  one 
liter  a  number  of  grams  equal  to  the  chemical  equivalent  (atomic  or  molec- 
ular weight  divided  by  the  valency).  A  solution  with  the  prefix  o.iw 
has  one-tenth  this  concentration,  etc.  For  example,  o.iwHCl  has  3.65 
grs.  of  HC1  (gas)  in  one  liter  of  solution,  or  that  proportion. 


240 


TABLES. 

TABLE  XXIV. 
Dielectric    Constants. 


I 

II 

Hydrocyanic  acid  ..... 

06 

Ether 

4^ 

Water  . 

80 

Xylol 

2     ?fi 

Methyl  alcohol  

•2  ? 

Benzol  

2   2 

Ethyl    alcohol  

2  C 

Toluol  

2   2 

Ammonia  (liquid)  

22 

Petroleum   

2.  07 

Acetone  
Sulphur    dioxide  .... 

17 
14. 

Pyridene  

1  2 

NDEX. 


Abbe  f ocometer,  132 
Aberration,  129 
Absorption,  electric,  185 
Acceleration  of  gravity,  33 
Air,  density  of,  30 

thermometer,  71 
Alloys,  melting-point,  loo 
Alternating    current    measurements, 

213,  214 

Ammeter,  calibration  of,  187,  208 
Anderson's  method   (self  induction), 

193 

Angle  of  prism,  123 

of  contact,  229 
Angular  field  of  view,  133 
Apparent  expansion  of  gas,  71 

of  liquid,  69 
Arc  of  vibration,  correction,  226 


Balance,  20-23 

correction  for  air  buoyancy,  22 

method  of  oscillations,  21-22 

ratio  of  arms,  22 

Ballistic  galvanometer,  147,  151,  197, 
Barometer,  19 

table  of  corrections,  226 
Battery,    electromotive    force,     181, 
183,  185 

resistance,  175,  183 
Beckman  thermometer,  6 1 
Biquartz,  142 
Bismuth  spiral,  196 
Boiling-point  of  water  (table),  233 
Bridge,  Carey  Foster,  173 

Thomson's  double,  172 

Wheatstone's,  145,  161 
Bunsen  photometer,  118 


Cadmium  cell,  153 
Calibrating  coil,  197,  201 
Calibration  of  ammeter,  187,  208 

of  galvanometer,  170,  197 

of  resistances,  173 


Calibration  of  scale,  24 

of  thermometer,  62-66 

of  voltmeter,  185 

Callender's  equation  (platinum  ther- 
mometer), 109 
Calorimeter,  for  gases,  105 

for  liquids,  106 

for  solids,  102,  103 

simple,  78 

Candle-power,  measurement  of,  120 
Capacity,  absolute  measurement,  192 

divided  charge  method,  191 

measurement    (alternating    cur- 
rents), 214,  215 
Capacities,  comparison  of,  189 

different  types,  184,  217 
Carey  Foster  bridge,  173 
Cathetometer,  17 
Chemical  hygrometer,  78 
Chromatic  aberration,  129 
Clark  cell,  152 

Clement  and  Desermes'  method  (spe- 
cific heat  of  gases),  84 
Coefficient  of  apparent  expansion,  69 

of  expansion,  66,  227,  228 

of  friction,  37-41 

of  increase  of  pressure,  71 

of  mutual  induction,  195,  216 

of  self  induction,  192 

of  viscosity,  51,  229 
Coincidence  method,  35 
Commutator,  double,  153 
Comparator,  14 
Condenser,  see  capacity. 
Conductivity,  thermal,  94 

of  electrolyte,  178 
Copper  voltameter,  208 
Chromatic  aberration,  129 
Current,  measurement  of,  187,  208 
Curves,  plotting  of,  10 

Daniell  cell,  152 
Demagnetization  of  iron,  204 
Density,  of  gases,  30,  226 
of  liquids,  27 


241 


242 


INDEX. 


Density,  of  powders,  29 

of  solids,  26,  227,  228 

of  water,  226 
Dew-point  77 
Dielectric  constant,  216 

(table),  240 

Diffraction  grating,  137 
Dip  circle,  158 
Dividing  engine,  16 
Dolezalek  electrometer,  167 
Double  bridge,  172 
Double  commutator,  153 
Drude's  apparatus   (electric  waves), 
218 

Earth  inductor,  160 

Elastic  constants,  (table),  228 

Electric  absorption,  185 

furnace,  107 
Electrical  resonance,  218-221 

units,  154 

waves,  218 

Electrolytes,  resistance  of,  178 
Electrometer,  quadrant,  167 
Electromotive  force,  device  for  small, 

153 

measurement  of,  181,  183,  185 

of  various  cells,  152-153 
Equivalent,  chemical,  28 
Errors,  2-10 

of  weights,  25 
Expansion,  apparent,  69 

coefficient  of,  69,  227,  228 
Eutectic  alloy,  101 

Focal  length  of  lenses,  128,  131 

of  mirrors,  127 
Focometer,  132 

Frequency  of  tuning  fork,  40,  112 
Friction,  coefficient  of  kinetic,  38 

coefficient  of  static,  37 
Fusion,  latent  heat  of,  87 

"G, "  determination  of,  33 
Galvanometer,  145-152 

ballistic,  147,  157 

bringing  to  rest,  149 

calibration  of,  170,  197 

damping,  149 

d'Arsonval,  146,  148 

different  types,  145 

flux,  147,  148 

resistance  of,  163,  164 

shunt,  150-152,  163 

study  of  ballistic,  147,  151,  197 

tangent,  209 

Thomson,  146 


Gas,  coefficient  of  increase  of  pressure, 

71 

density  of,  30 

heat  value,  105 

ratio  of  specific  heats,  84,   113, 

115,  229 
Grating,  diffraction,  137 

Half  shade  polarimeter,  143 
Heat,  conductivity  for,  94 
Heat  value  of  gas,  105 

mechanical  equivalent,  97,  207 

of  liquid,  106 

of  solid,  102,  103 
Hempel  calorimeter,  102 
Hooke's  law,  42 
Horizontal  component  of  earth's  field, 

154,  207 

Hygrometry,  77,  232 
Hypsometer,  65 
Hysteresis,  203 

Incandescent  lamp,  study  of,  120 

Inclination,  magnetic,  158 

Index  of  refraction,  measurement  of, 

124 

table  of,  237 
Inertia,  measurement  of  moment  of, 

48 

Insulation  resistance,  167 
Interferometer,  139 
Iron,  permeability  of,  199 

Junker  calorimeter,  105 

Kundt's  method  (velocity  of  sound), 
114 

Latent  heat  of  fusion,  87 

of  vaporization,  89,  92 
Lenses,  combinations,  131 

focal  length,  128,  131 

radii  of  curvature,  125 

rule  of  signs,  117 
Light,  filters,  116 

monochromatic,  116 

wave  length,  137,  234 
Logarithmic  decrement,  149 

tables,  224,  225 
Low  resistance,  measurement  of,  169- 

173 
Lummer-Brodhun  photometer,  118 

Magnetic  field,  measurement  of,  196 
of  earth,  dip,  158 
of  earth,  horizontal    component, 
154,  207 


INDEX. 


243 


Magnetic  hysteresis,  203 

permeability,  199 
Magnetometer,  156 
Magnification,  132 
Magnifying  power  of  telescope,  133 
Mance's  method  (battery  resistance), 

175 

Mean  deviation,  4 

Mechanical  equivalent  of  heat,  elec- 
trical method,  207 
by  friction,  97 
Melting-point  of  alloy,  100 
of  metals  (table),  235 
Mercury,  vapor  pressure   of    (table), 

235 

Michelson's  interferometer,  139 
Micrometer  caliper,  13 

microscope,  13 
Minimum  deviation,  124 
Mirror  and  scale,  adjustment  of,  23 
Mirrors,    spherical,    measurement   of 
focal  length,  125 

rule  of  signs,  117 
Moduli,  law  of,  28 
Mohr-Westphal  balance,  27 
Moment  of  inertia,  48 
Monochromatic  light,  116 
Mutual  induction,  195,  216 

Optical  lever,  44,  67 
pyrometer,  108 


Passages,  method  of,  49 
Pendulum,  correction  for  arc  (table), 
227 

physical,  33 

simple,  33 
Permeability,  199 
Photometric  table,  238 
Photometry,  118 
Pirani's  method  (mutual  induction), 

195 

Pitch  of  tuning  fork,  40,  112 
Planimeter,  206 
Platinum  thermometer,  107 
Pohl  commutator,  205 
Polarization,  rotation  of  plane  of,  140 
Possible  error,  4-9 
Post-office  box  bridge,  145 
Polarimeter,  biquartz,  142 

half  shade,  143 
Potential  difference,  measurement  of, 

185 

Potentiometer,  187 
Pressure,  coefficient  of  increase  of,  71 
of  mercury  vapor,  233 


Pressure  of   water  vapor    (measure- 
ment), 74,  (table),  232 
Primus  burner,  106 
Prism,  angle  of,  123 

minimum  deviation,  124 
Probable  error,  9 
Pyknometer,  30 
Pyrometry,  106,  108 

Quadrant  electrometer,  167 

Radiation  correction,  59-61 

pyrometer,  108 

Radius  of  curvature  of  mirror,  125 
Ratio  of  specific  heats,  measurement 
of,  84,  113,  115 

table,  230 
Refractive  index,  measurement  of,  124 

of  lenses,  127 

table,  237 
Regnault's  apparatus,  hygrometry,  77 

vapor  pressure,  75 
Reports,  2 
Resistance,  boxes,  144 

comparison  of,  173 

electrolytic,  178 

high,  166-169 

low,  169-173 

measurement  of,  161,  213 

of  ballistic  galvanometer,  197 

of  battery,  175,  184 

of  galvanometer,  163,  164 

specific,  162,  176,  237 

temperature  coefficient  of,  176 
Resistances,  comparison  of,  173 
Resolving  power,  of  eye,  136 

of  telescope,  135 
Rigidity  of  metals,  47 
Rosenhain  calorimeter,  103 
Rotation    of    plane    of    polarization, 
measurement  of,  140 

table,  237 
Rubber  grease,  29 

Saccharimetry,  142 
Scale,  calibration  of,  24 

construction  of,  24 
Self    induction,    alternating    current 
method,  213,  215 

Anderson's  method,  192 

comparison  of,  194 
Shunts,  galvanometer,  150-152,  163 
Shear  modulus,  47 
Signs  (mirrors  and  lenses),  117 
Slfde  wire  bridge,  145 
Sound  velocity  of,  in,  114 


244 


INDEX. 


Specific  gravity  bottle,  30 
heat,  of  gases,  84 

(table),  230 
of  metals,  78 
(table),  229 
of      miscellaneous       substances 

(table),  229,  238 
inductive  capacity,  see  dielectric 

constant, 
resistance  of  electrolytes,  178,  237 

of  metals,  162,  176,  237 
rotatory  power,  140 

(table),  237 
Spectrometer,  122 
Spherical  aberration,  129 
Spherometer,  14 
Standard  cells,  152 
Stroboscopic  disk,  40 
Surface  tension,  measurement  of,  57 
table,  231 

Tangent  galvanometer,  209 
Telescope,  adjustment  of,  23 

magnifying  power  of,  133 

resolving  power  of,  135 
Temperature  coefficient  of  expansion, 
66 

of  expansion  (tables),  228-230 

of  resistance,  176 
(table),  239 

of  surface  tension,  229 
Thermal  conductivity,  94 
Thermocouple,  87,  107,  211 
Thermometer,  air,  71 

Beckman,  61 

calibration  of,  62-66 

fixed  points,  64 

platinum,  107 
Thomson's  double  bridge,  172 

galvanometer,   146 

method      (galvanometer    resist- 
ance),  164 

method  of  mixtures,  191 
Time  of  vibration,  method  of  coin- 
cidences, 35 

method  of  passages,  49 

reduction  to  infinitely  small  arc, 
226 


Time  of  vibration,  signals,  23 
Torsion,  modulus  of,  47 
Trigonometrical  functions  (table),  226 
Tuning  fork,  pitch  of,  40,  112 

Units,  electrical,  154 

Vacuum,  reduction  of  weighing  to,  22 

Valson's  law  of  moduli,  28 

Vapor  pressure  of  mercury   (table), 

235 

of  water  (measurement),  74 

(table),  232 

Vaporization,  latent  heat  of,  89,  92 
Velocity  of  sound,  Kundt's  method, 
114 

resonance  method,  in 
Vernier,  12 

caliper,  13 

Virtual  image,  127,  130 
Viscosity,  measurement  of  coefficient 
of,  51 

table,  229 

Voltameter,  copper,  208 
Voltmeter,  calibration,  185 
Volumenometer,  29 
Water,  boiling-point  (table),  233 

density  (table),  228 

equivalent,  82 

vapor  pressure,  74,  230 
Wave  length,  of  electric  waves,  218 

of   light    waves    (measurement), 

137 
(table),  236 

of  sound  waves,  in,  114 
Weighing,  by  oscillations,  21 

double,  22 

reduction  to  vacuum,  22 
Weight  thermometer,  69 
Weights,  calibration  of,  25 
Weston  (cadmium)  cell,  153 
Wet  and  dry  bulb  hygrometer,  78,232 
Wheatstone's  bridge,  144,  161 


Young's  modulus,  by  bending,  43 
by  stretching,  42 
table,  228 


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